# A Trivial Geometrical Phase of an Electron Wavefunction in a Direct Band Gap
Semiconductor CdGeAs2

Vikas Saini<EMAIL_ADDRESS><EMAIL_ADDRESS>Souvik Sasmal
Vikash Sharma  Suman Nandi  Gourav Dwari  Bishal Maity  Ruta Kulkarni
Arumugam Thamizhavel  Department of Condensed Matter Physics and Materials
Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba,
Mumbai 400 005, India.

###### Abstract

Chalcopyrite compounds are extensively explored for their exotic topological
phases and associated phenomena in a variety of experiments. Here, we discuss
the electrical transport properties of a direct energy gap semiconductor
CdGeAs2. The observed transverse magnetoresistance (MR) is found to be around
136$\%$ at a temperature of 1.8 K and a magnetic field of 14 T, following the
semiclassical exponent MR $\sim$ B2.18. The MR analysis exhibits a violation
of the Kohler rule, suggesting the involvement of multiple carriers in the
system. Below 15 K, with decreasing magnetic field, the MR increases, leading
to the well known quantum interference phenomenon weak localization (WL). The
analysis of the magnetoconductivity data based on the Hikami-Larkin-Nagaoka
(HLN) model unveils three dimensional nature of the WL and the weak spin-orbit
coupling in CdGeAs2. The phase coherence length follows the Lϕ $\sim$
$T^{-0.66}$ power law, which exhibits the 3D nature of the observed WL
feature.

††preprint: AIP/123-QED

In condensed matter physics the electronic phases are classified primarily
based on their band gap and order of the carrier densities that lead to many
phenomena and novel properties that are driven by symmetry breaking of the
systems [1, 2, 3]. On the contrary, topological phases of materials exhibit
symmetry protected phases along with the combination of novel band crossings.
The topological materials lead to a plethora of interesting properties like
quantum Hall effect, quantum spin Hall effect, linear magnetoresistance and
weak antilocalization, ultrahigh magnetoresistance, chiral anomaly, etc. which
have been experimentally measured on the number of materials [4, 5, 6, 7, 8].
However, quantum interference phenomena such as weak localization and weak
antilocalization are also observed in the semiconducting compounds which have
been explained thoroughly by the existing theories [9, 10].

Chalcopyrite materials crystallize in the general chemical formula $ABC_{2}$
where $C$ represents the pnictide elements, are being studied extensively for
their novel features to transport experiments. Many of the compounds of this
series have been predicted and realized experimentally which appear in the
whole range of the electronic phases of the band inversion to double band
inversions. The ideal Weyl phase have been proposed by Ruan et al, for CuTlSe2
and ZnPbAs2, etc [11].

Thermoelectric properties of this series of semiconducting materials are
crucial due to the sharp density of states near the Fermi level, as studied in
CuGaTe2 and Ag1-xGaTe2 materials [12, 13, 14]. These materials exhibit large
thermopower.

Here, we discuss the transport properties of a chalcopyrite material CdGeAs2.
The temperature dependence of resistivity shows a negative slope across the
entire temperature range from 1.8 to 300 K, indicating the semiconducting
nature of this compound. The MR is positive for temperatures above around 15 K
in the magnetic field range from -14 to +14 T. At lower temperatures (less
than 15 K), the MR increases as the magnetic field decreases, indicating the
weak localization phenomena in CdGeAs2. The estimated small value of $\alpha$
suggests weak spin-orbit coupling (SOC) in CdGeAs2.

We also examined the thermopower ($S$) of CdGeAs2 above room temperature. The
negative thermopower indicates the dominant $n$-type carriers in the system,
which intrinsically appear in the crystal growth process and are not due to
any external doping. High values of $S$ are observed for the B1 and B2 samples
at the temperatures around 530 K with the values of -185 and -285 $\mu$V/K,
respectively.

The material CdGeAs2 was prepared using the modified Bridgman method by
involving fast heating and natural cooling of the sample. Initially, the
stoichiometric ratio of the individual elements Cd, Ge, and As was taken in an
alumina crucible, which was then sealed inside a quartz tube under a vacuum of
around 10-5 mbar. The tube was placed in a box furnace at a temperature of
around 500 ∘C for 4 days later the furnace was switched off for natural
cooling. The tiny pieces of polycrystals of CdGeAs2 were obtained, and a small
amount of crystals were isolated and crushed into powders for powder X-ray
diffraction.

Before performing electrical and thermal transport measurements, pellets of
CdGeAs2 were made by applying $\sim$ 5 GPa of hydrostatic pressure for nearly
5 hrs on the fine powders of CdGeAs2 using tungsten carbide die. The pellets
were then sealed under vacuum inside the quartz tube and kept at 400 ∘C for 3
days for the sintering process. The electrical resistivity measurements were
performed using the conventional four probe method, where the electrical
contacts were made by the two component silver paste with gold wires attached
to the sample surface. A 1.28$\times$2.68$\times$0.21 mm3 dimensions sample
was used for the resistivity measurements. The electrical transport
measurements were carried out in a PPMS (Quantum Design, USA) equipped with a
14 T magnetic field, over a temperature range of 1.8 - 300 K. Thermopower was
measured in a homemade setup where the sample was placed between the two
copper plates for the thermal gradient. The experimental setup is described in
reference [15].

Thermopowers were calculated using the Boltzmann transport equation under the
constant scattering time approximation (CSTA), as implemented in BoltzTraP2.
The calculations were based on self-consistent results from the full potential
linearized augmented plane wave (FP-LAPW) method implemented in WIEN2k with
10,000 $k$ points. The thermopower results were converged using 149,784
irreducible $k$ points [16, 17, 18, 19, 20].

Figure 1: (a) Crystal structure of CdGeAs2. The red, green, and purple spheres
are Cd, Ge, and As atoms, respectively. (b) The powder X-ray diffraction
pattern of CdGeAs2, where red open circles correspond to the experimentally
observed pattern, the black line is the refined rietveld fit, and blue ticks
represent the peak positions. Figure 2: (a) Longitudinal resistivity as a
function of temperature in applications of various static magnetic fields. (b)
Thermally excited band gap as a function of the magnetic field up to 14 T. (c)
Magnetoresistance (MR) versus magnetic field plot for several temperatures
from 1.8 to 200 K. (d) MR plot at $T$ = 1.8 K up to 14 T, where $B^{n}$ is
fitted with $n$ = 2.18. (e) MR dependence with the temperatures at 14 T
magnetic field. (f) Kohler plot at various temperatures with exponent $n$ =
2.18. Figure 3: (a) MR plot as a function of the magnetic field, where the
lower regime of magnetic field exhibits negative MR where WL is observed. (b)
Modified HLN fitted normalized electrical conductivity with a magnetic field.
(c) Phase coherence length ($L_{\phi}$) as a function of temperature, where
the red plot is the temperature exponent fit. (d) Obtained HLN fitting
parameter $\alpha$ for various temperatures.

CdGeAs2 forms a tetragonal unit cell where CdGe layers are sandwiched between
As layers along the longest crystal axis. A single unit cell consists of 9
layers. The bottom and top four layers are mirror reflections of each other in
the CdGe (5th) layer, combined with a 90∘ rotation depicts an inversion
asymmetry in CdGeAs2. However, CdGeAs2 falls into the non-symmorphic space
group category I$\bar{4}$2d (No. 122), possessing lattice translation symmetry
combined with other crystalline symmetries. The atoms in a unit cell are
diagonally mirror reflected in the $\pm$mxy planes, whereas mx and my mirror
symmetries are absent in CdGeAs2 (see Fig. 1(a)).

The powder X-ray diffraction pattern is shown in the Fig. 1(b). Experimentally
observed XRD peaks match with the Rietveld refined pattern that depicts the
phase purity of the prepared CdGeAs2 sample. The refined pattern for CdGeAs2
is depicted in the black color and the estimated lattice parameters are $a$ =
$b$ = 5.941 $\rm\AA$, and $c$ = 11.214 $\rm\AA$.

We shall discuss the electrical transport results on CdGeAs2 polycrystalline
sample. The resistivity $\rho$(T) of CdGeAs2 is found to be small, nearly
0.019 $\Omega$-m at 300 K, and it starts to increase with decreasing
temperature, attaining the value of 38 $\Omega$-m at 1.8 K. The slope of the
resistivity in the entire temperature range is negative
($\frac{d\rho(T)}{dT}<0$), which leads to the conclusion of the semiconducting
nature of CdGeAs2. In addition, the semiconducting phase of this material has
been optically studied, where a direct band gap of around 0.57 eV at room
temperature is observed, which further increases with lower temperatures.
First principles calculations using modified Becke Johnson (mBJ) exchange
potential to exhibit a direct band gap semiconductor of 0.65 eV gap,
consistent with the experimental results. Thus, it supports establishing the
semiconducting phase in CdGeAs2 [21, 22, 23].

The electrical resistivity measured in the presence of static magnetic fields
up to 14 T is depicted in Fig. 2(a). The resistivity below around 150 K starts
to increase significantly with increasing magnetic field. The thermal
activation gap is estimated using the Arrhenius equation
$\sigma\sim\sigma_{0}$ exp(-E${g}$/k${B}$T), where $k_{\rm B}$ is the
Boltzmann constant, $E_{\rm g}$ is the band gap, and $T$ is the absolute
temperature. The thermally excited band gap is around 98.6 meV in a zero
magnetic field, which is roughly 1.5 times more than the band gap value at
room temperature. The application of a magnetic field slowly increases $E_{g}$
linearly with an increase in the magnetic field, reaching nearly 101.3 meV for
a 14 T magnetic field, as shown in Fig. 2(b).

The magnetoresistance( MR($\%$) =
$\frac{\rho_{xx}(B)-\rho_{xx}(0)}{\rho_{xx}(0)}$$\times$100) of CdGeAs2
increases with the increasing magnetic field for every temperature from around
15 to 300 K as shown in Fig. 2(c). For lower temperatures less than 15 K MR
becomes negative in low field regime owing to the quantum interference
phenomenon of weak localization and this feature is suppressed with increasing
magnetic fields as we discuss this phenomenon later in detail. The observed
value of MR at 14 T and 1.8 K is 136$\%$ and an increase in temperature also
increases MR until $\sim$ 50 K that reaches up to 154$\%$. Beyond this
temperature, it reduces rapidly down to 55$\%$ at 150 K and then slowly
decreases until $T$ = 300 K.

According to the semiclassical Boltzmann theory, in the weak field limit
($\omega\tau<<1$), the MR follows a quadratic field dependence, and for the
strong field limit, it gets saturated.

However, we do not observe saturation in MR up to an applied magnetic field of
14 T. Therefore, we can use a semiclassical description of the unsaturated MR
by employing MR $\sim$ Bn, where $n$ = 2.18 which is shown in the Fig. 2(c).

We observed that the MR plots for various temperatures do not overlap onto
each other when plotted against ($\frac{B}{\rho_{0}}$)n, which implies the
violation of the Kohler rule. This suggests the presence of multiple carriers
in CdGeAs2. The Kohler plot is depicted in the Fig. 2(f) in the temperature
range 1.8 - 20 K, where $n$ = 2.18 is the obtained exponent of MR.

The low magnetic field data up to a temperature of 10 K is shown in Fig. 3(a),
which reveals negative MR down to 1.12$\%$ for $T$ = 1.8 K. At lower magnetic
fields, the MR increases, suggesting the weak localization (WL) in CdGeAs2.
However, this effect disappears at temperatures above 15 K and the MR becomes
positive in the entire range of the magnetic field.

We observed WL phenomenon in the CdGeAs2. As we mentioned earlier that from
the semiclassical theory MR increases quadratically in the weak limit and it
gets saturated in the strong fields. But the quantum corrections in the low
field regime leads to the quantum interference phenomena as WL and WAL
depending on the interference between the time reversal invariant paths of
electron wavefunctions as widely observed for the number of crystalline
systems. For this quantum phenomenon, the phase coherence length ($L_{\phi}$)
must be greater than the mean free path ($l$) that results in the interference
between time reversal (TR) paths of the electron wavefunctions that
essentially change the electrical conductivity of the measured compounds. If
the phase difference between the TR paths is constructive then it results in
more backscatterings in the system thus leads to the suppression of electrical
conductivity from the normal classical behavior that is known as the WL. On
the other hand, destructive interference gives a $\pi$ phase between the TR
paths that suppresses the backscatterings in the system and hence leads to an
increase in electrical conductivity that is defined as WAL. The increase in
temperature causes a decrease in $L_{\phi}$ which essentially suppresses these
features. This whole phenomenon appears in the time reversal invariant
condition therefore, it is observed in the lower magnetic field regime and the
application of the high magnetic field suppresses this feature.

To observe WAL, the strong spin-orbit coupling is required to induce a
nontrivial phase that increases electrical conductivity. This effect can be
two- or three-dimensional in nature. The Hikami-Larkin-Nagaoka (HLN) model is
commonly used to explain WL and WAL in two dimensional materials and has been
applied to identify surface states in topological insulators [8, 24].

To understand the weak localization in CdGeAs2 we employed the HLN model to
analyze the experimental data that alone does not produce the good fittings
with the experimental data of normalized magnetoconductivity ($\Delta$$\sigma$
= $1/\rho$(B) - $1/\rho$(0)); where $\rho$ is the electrical resistivity.
However, the fit was not good. This suggested classical contributions of the
cyclotron orbits should also be taken into account. Hence we added the
additional $B^{2}$ term to obtain a reasonably good fit of the
magnetoconductivity data. Further, we have incorporated the three dimensional
term of the weak localization $B^{1/2}$ which also improved the HLN fitting.
Our estimated $\alpha$ parameter from the modified HLN model is around 8 m-1
at 1.8 K which is reasonably lower than the observed values in the three
dimensional nature of LuPdBi and LuPtSb [25, 26]. However, the lower value of
$\alpha$ could be attributed to the semiconducting nature of the material and
observed WL suggests the low spin-orbit coupling strength in CdGeAs2 as
observed in the annealed semiconducting material PbPdO2 [27].

The fitted normalized magnetoconductivity $\Delta\sigma$ is depicted in the
Fig. 3(b) for various temperatures up to 10 K. The modified HLN equation is
expressed as follows [28, 29].

$\Delta\sigma(B)=\frac{\alpha
e^{2}}{2\pi^{2}\hbar}[\Psi(\frac{1}{2}+\frac{\hbar}{4eBL_{\phi}^{2}})-ln(\frac{\hbar}{4eBL_{\phi}^{2}})]-\beta
B^{2}+\delta B^{1/2}$ (1)

The obtained phase coherence length $L_{\phi}$ and $\alpha$ parameters are
shown in the Fig. 3(c-d). The phase coherence length relation with temperature
can be expressed as follows: $\frac{1}{L^{2}_{\phi}(T)}$ =
$\frac{1}{L^{2}_{\phi}(0)}$ \+ AeeT${}^{p^{\prime}}$ \+
AepT${}^{p^{\prime\prime}}$, where constants Aee and $A_{ep}$ reflects the
electron electron and electron phonon interactions. For the electron-electron
interaction dominating system exponent $p^{\prime}$ possesses 1 and 3/2 values
for the two and three-dimensional nature of WL, respectively. Since in CdGeAs2
the electron-electron interactions are dominant compared to the phonon-phonon
interactions, the above expression can be simplified as Lϕ $\sim$ T-p where
$p=p^{\prime}/2$. The temperature dependence of the $L_{\phi}$ results in the
exponent $p$ = 0.66 (Fig. 3(c)), which is close to the expected value of
$p=p^{\prime}/2=0.75$ suggesting the three-dimensional nature of the weak
localization.

The thermopower ($S$) of CdGeAs2 was measured on two samples B1 and B2 from
temperature $\sim$300 to 570 K. The magnitude of $S$ increases with
temperature, as shown in Fig.4, and the negative values indicate natural
$n$-type doping in CdGeAs2. The qualitative behavior of both samples is
similar, with a subtle shift in values that may be due to differences in
carrier densities. As shown in Fig. 4, the calculated $S$ for three different
carrier densities reveals that increasing carrier density leads to a decrease
in $|S|$, thus the carrier density of sample B2 should be smaller than that of
sample B1. The magnitude of $S$ observed falls within the range predicted for
$n$-type dopings. The thermopower calculations are performed under the
constant relaxation time approximation (CSTA) [20]. The WL phenomena is
generally related to the impurities/disorders present in the sample. Here in
this case of CdGeAs2 the observed WL is attributed to the
disorders/impurities. Further, it has to be pointed out that the possible
scatterings from the impurities near the Fermi level are not considered in
calculations therefore the difference between observed and calculated
thermopower is expected for CdGeAs2.

The effects of impurities and imperfections in crystal on observed thermopower
can be understood with the Mott relation [30] given in the Eq. 2

$S(n,T)=\frac{\pi^{2}k_{B}^{2}T}{3q}\left[\frac{1}{n}\frac{dn(E)}{dE}+\frac{1}{\mu}\frac{d\mu(E)}{dE}\right]_{E_{F}}$
(2)

Here $k_{B}$ is the Boltzmann constant and $q$ denotes the charge of carriers,
n(E) and $\mu$(E) represents the carrier density and mobility of the charge
carriers. $T$ stands for the absolute temperature. The first term in the Eq. 2
represents the contributions from the density of states at the Fermi level,
the sharp increase in density of states against energy leads to the large
thermopower whereas the second term appears from the scatterings caused by
disorders that have not been attributed in the calculated thermopower. It is
likely that the nonzero second term in Mott relation may attribute to the
difference between the calculated and experimentally measured thermopowers on
CdGeAs2.

Figure 4: Thermopower ($S$) is plotted against the temperature from 200 - 600
K. Red, green, and blue curves are the calculated $S$ for the different
carrier densities. Open and closed black circles are the experimentally
observed $S$ for two samples.

In conclusion, we studied the transport properties of a semiconducting
chalcopyrite system CdGeAs2 which exhibits weak localization phenomenon. For
temperatures below $\sim$150 K, semiclassical dependence of the
magnetoresistance is observed. The phase coherence length ($L_{\phi}$)
exponent with temperatures results in the three-dimensional nature of the WL
in CdGeAs2. Additionally, a large thermopower is measured for CdGeAs2 as a
consequence of the sharp increase in the density of states.

The authors gratefully acknowledge high temperature thermoelectric facility at
IIT Mandi, India for the thermopower measurements. VS and AT would like to
specially thank Prof. Sudhir Kumar Pandey and MR. Shamim Sk for their
assistance in the thermopower measurements.

## Data Availability Statement

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## References

## References

  * Arodz, Dziarmaga, and Zurek [2003] H. Arodz, J. Dziarmaga, and W. H. Zurek, Vol. 127 (Springer Science & Business Media, 2003).
  * Beekman, Rademaker, and van Wezel [2019] A. Beekman, L. Rademaker, and J. van Wezel, SciPost Phys. Lect. Notes , 011 (2019).
  * McGreevy [2022] J. McGreevy, arXiv preprint arXiv:2204.03045 (2022).
  * Liang  _et al._ [2015] T. Liang, Q. Gibson, M. N. Ali, M. Liu, R. Cava, and N. Ong, Nat. Mater. 14, 280–284 (2015).
  * Shekhar  _et al._ [2015] C. Shekhar, A. K. Nayak, Y. Sun, M. Schmidt, M. Nicklas, I. Leermakers, U. Zeitler, Y. Skourski, J. Wosnitza, Z. Liu, _et al._ , Nat. Phys. 11, 645–649 (2015).
  * Feng  _et al._ [2015] J. Feng, Y. Pang, D. Wu, Z. Wang, H. Weng, J. Li, X. Dai, Z. Fang, Y. Shi, and L. Lu, Phys. Rev. B 92, 081306 (2015).
  * Saini  _et al._ [2021] V. Saini, S. Sasmal, R. Kulkarni, and A. Thamizhavel, Appl. Phys. Lett. 119, 071904 (2021).
  * Lu and Shen [2014] H.-Z. Lu and S.-Q. Shen, in  _Spintronics Vii_ , Vol. 9167 (SPIE, 2014) pp. 263–273.
  * Newton  _et al._ [2017] P. Newton, R. Mansell, S. Holmes, M. Myronov, and C. Barnes, Appl. Phys. Lett. 110, 062101 (2017).
  * Kilanski  _et al._ [2013] L. Kilanski, K. Szałowski, R. Szymczak, M. Górska, E. Dynowska, P. Aleshkevych, A. Podgórni, A. Avdonin, W. Dobrowolski, I. Fedorchenko, _et al._ , J. Appl. Phys. 114, 093908 (2013).
  * Ruan  _et al._ [2016] J. Ruan, S.-K. Jian, D. Zhang, H. Yao, H. Zhang, S.-C. Zhang, and D. Xing, Phys. Rev. Lett. 116, 226801 (2016).
  * Kumar Gudelli  _et al._ [2013] V. Kumar Gudelli, V. Kanchana, G. Vaitheeswaran, A. Svane, and N. E. Christensen, J. Appl. Phys. 114, 223707 (2013).
  * Yusufu  _et al._ [2011] A. Yusufu, K. Kurosaki, A. Kosuga, T. Sugahara, Y. Ohishi, H. Muta, and S. Yamanaka, Appl. Phys. Lett. 99, 061902 (2011).
  * Parker and Singh [2012] D. Parker and D. J. Singh, Phys. Rev. B 85, 125209 (2012).
  * Sk, Pandey, and Pandey [2022] S. Sk, A. Pandey, and S. K. Pandey, “Instrument for simultaneous measurement of seebeck coefficient and thermal conductivity in the temperature range 300–800 k with python interfacing,” Rev. Sci. Instrum. 93, 043902 (2022).
  * Blaha  _et al._ [2001] P. Blaha, K. Schwarz, G. K. Madsen, D. Kvasnicka, J. Luitz, _et al._ , An augmented plane wave+ local orbitals program for calculating crystal properties (2001).
  * Schwarz and Blaha [2003] K. Schwarz and P. Blaha, Comput. Mater. Sci. 28, 259–273 (2003).
  * Schwarz [2003] K. Schwarz, J. Solid State Chem. 176, 319–328 (2003).
  * Madsen and Singh [2006] G. K. Madsen and D. J. Singh, Comput. Phys. Commun. 175, 67–71 (2006).
  * Madsen, Carrete, and Verstraete [2018] G. K. Madsen, J. Carrete, and M. J. Verstraete, Comput. Phys. Commun. 231, 140–145 (2018).
  * McCrae  _et al._ [1997] J. McCrae, R. Hengehold, Y. Yeo, M. Ohmer, and P. Schunemann, Appl. Phys. Lett. 70, 455–457 (1997).
  * Akimchenko, Ivanov, and Borshchevsky [1973] I. Akimchenko, V. Ivanov, and A. Borshchevsky, Sov. Phys. Semiconduct. 7, 309–310 (1973).
  * Bai  _et al._ [2005] L. Bai, C. Xu, P. Schunemann, K. Nagashio, R. Feigelson, and N. Giles, J. Phys. Condens. Matter 17, 549 (2005).
  * Liu  _et al._ [2012] M. Liu, J. Zhang, C.-Z. Chang, Z. Zhang, X. Feng, K. Li, K. He, L.-l. Wang, X. Chen, X. Dai, Z. Fang, Q.-K. Xue, X. Ma, and Y. Wang, Phys. Rev. Lett. 108, 036805 (2012).
  * Xu  _et al._ [2014] G. Xu, W. Wang, X. Zhang, Y. Du, E. Liu, S. Wang, G. Wu, Z. Liu, and X. X. Zhang, Sci. Rep. 4, 5709 (2014).
  * Hou  _et al._ [2015] Z. Hou, Y. Wang, G. Xu, X. Zhang, E. Liu, W. Wang, Z. Liu, X. Xi, W. Wang, and G. Wu, Appl. Phys. Lett. 106, 102102 (2015).
  * Choo  _et al._ [2015] S. Choo, K. Lee, S. Park, J. Yoon, G. Park, C.-Y. You, and M. Jung, Appl. Phys. Lett. 106, 172404 (2015).
  * Assaf  _et al._ [2013] B. A. Assaf, T. Cardinal, P. Wei, F. Katmis, J. S. Moodera, and D. Heiman, Appl. Phys. Lett. 102, 012102 (2013).
  * Wang  _et al._ [2020] J. Wang, X. Liu, C. Bunker, L. Riney, B. Qing, S. Bac, M. Zhukovskyi, T. Orlova, S. Rouvimov, M. Dobrowolska, _et al._ , Phys. Rev. B 102, 155307 (2020).
  * He and Tritt [2017] J. He and T. M. Tritt, Sci. 357, eaak9997 (2017).
  * Hu and Rosenbaum [2008] J. Hu and T. Rosenbaum, Nat. Mater. 7, 697–700 (2008).
  * Yang  _et al._ [1999] F. Yang, K. Liu, K. Hong, D. Reich, P. Searson, and C. Chien, Sci. 284, 1335–1337 (1999).
  * Hu, Parish, and Rosenbaum [2007] J. Hu, M. M. Parish, and T. Rosenbaum, Phys. Rev. B 75, 214203 (2007).
  * Mitra  _et al._ [2019] S. Mitra, J. G. S. Kang, J. Shin, J. Q. Ng, S. S. Sunku, T. Kong, P. C. Canfield, B. S. Shastry, P. Sengupta, and C. Panagopoulos, Phys. Rev. B 99, 045119 (2019).
  * Narayanan  _et al._ [2015] A. Narayanan, M. Watson, S. Blake, N. Bruyant, L. Drigo, Y. Chen, D. Prabhakaran, B. Yan, C. Felser, T. Kong, _et al._ , Phys. Rev. Lett. 114, 117201 (2015).
  * Wang  _et al._ [2012] K. Wang, D. Graf, L. Wang, H. Lei, S. Tozer, C. Petrovic, _et al._ , Phys. Rev. B 85, 041101 (2012).
  * Willardson and Duga [1960] R. Willardson and J. Duga, Proc. Phys. Soc. (1958-1967) 75, 280 (1960).
  * Johnson  _et al._ [2010] H. Johnson, S. Bennett, R. Barua, L. Lewis, and D. Heiman, Phys. Rev. B 82, 085202 (2010).
  * Halbo and Sladek [1968] L. Halbo and R. Sladek, Phys. Rev. 173, 794 (1968).
  * Xu  _et al._ [1997] R. Xu, A. Husmann, T. Rosenbaum, M.-L. Saboungi, J. Enderby, and P. Littlewood, Nature 390, 57–60 (1997).
  * Hu, Rosenbaum, and Betts [2005] J. Hu, T. Rosenbaum, and J. Betts, Phys. Rev. Lett. 95, 186603 (2005).
  * Lee  _et al._ [2002] M. Lee, T. Rosenbaum, M.-L. Saboungi, and H. Schnyders, Phys. Rev. Lett. 88, 066602 (2002).
  * Abrikosov [1998] A. Abrikosov, Phys. Rev. B 58, 2788 (1998).
  * Yang  _et al._ [2016] X. Yang, H. Bai, Z. Wang, Y. Li, Q. Chen, J. Chen, Y. Li, C. Feng, Y. Zheng, and Z.-a. Xu, Appl. Phys. Lett. 108, 252401 (2016).
  * Parish and Littlewood [2003] M. Parish and P. Littlewood, Nature 426, 162–165 (2003).
  * Abrikosov [2000] A. Abrikosov, “Quantum linear magnetoresistance,” EPL 49, 789 (2000).
  * Pavlosiuk, Kaczorowski, and Wiśniewski [2015] O. Pavlosiuk, D. Kaczorowski, and P. Wiśniewski, Sci. Rep. 5, 1–9 (2015).


From Ramanujan to renormalization: the art of doing

away with divergences and arriving at physical results

Wolfgang Bietenholz

Instituto de Ciencias Nucleares

Universidad Nacional Autónoma de México

A.P. 70-543, C.P. 04510 Ciudad de México, Mexico

A century ago, Srinivasa Ramanujan — the great self-taught Indian genius of
mathematics — died, shortly after returning from Cambridge, UK, where he had
collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding
results to different branches of mathematics, like analysis and number theory,
with a focus on special functions and series. Here we refer to apparently
weird values which he assigned to two simple divergent series, $\sum_{n\geq
1}n$ and $\sum_{n\geq 1}n^{3}$. These values are sensible, however, as
analytic continuations, which correspond to Riemann’s $\zeta$-function.
Moreover, they have applications in physics: we discuss the vacuum energy of
the photon field, from which one can derive the Casimir force, which has been
experimentally measured. We further discuss its interpretation, which remains
controversial. This is a simple way to illustrate the concept of
renormalization, which is vital in quantum field theory.

Keywords: Ramanujan summation, Casimir effect, renormalization,
$\zeta$-function

##  1 Ramanujan’s letter from 1913

101 years ago, Srinivasa Ramanujan111Phonetically, his last name could be
written in Spanish as “Ramánuchan”. (1887-1920) passed away in Madras, at that
time part of the British Empire (since 1996, this state capital in South-East
India is named Chennai). He was one of the greatest geniuses in the history of
mathematics. One way to measure the impact of his work is through the amount
of mathematical terms that bear his name: the mathematical online encyclopedia
Wolfram Mathworld [1] documents 27 terms named after Ramanujan, and his name
appears in a total of 205 items; in both respects, he is among the leading
mathematicians of all times.222To be explicit, if we rank mathematicians by
the number of mathematical items named after them, Ramanujan is at position 6,
following Euler (71), Gauss (48), Hilbert (33), Fermat (32) and Riemann (31),
and followed by Cauchy (26), Dirichlet, Jacobi, Weierstraß (23 each), Euclid
(22) and Poincaré (21). Regarding the number of mentionings in a Mathworld
entry, Ramanujan is at position 18. This is particularly amazing since
Ramanujan started to elaborate stunning equations with hardly any mathematical
education,333Ramanujan only obtained from a friend a library copy of a book by
George Carr [2], which he studied intensively. It is a collection of formulae
and theorems, with little explanation, written as an overview for students who
are preparing for exams. and he died at the age of only 32 (younger than
Mozart, for example).

Figure 1: Srinivasa Ramanujan (1887–1920)

Ramanujan was born in 1887 in a town called Erode, but at the age of 2 his
mother took him to the city of Madras, some 400 km away. In the early 20th
century he lived in extreme poverty, at the edge of starvation, but he
discovered a multitude of important mathematical formulae, based on his
incredible intuition — I tend to interpret it as a kind of “pattern
recognition” (although it was not an automated process).

In 1912, he began to send letters to British mathematicians, trying to attract
attention to his discoveries; for a while without success. In January 1913, he
finally wrote to Godfrey Hardy, a brilliant young mathematician at Trinity
College of Cambridge University, who — together with his long-term
collaborator John Littlewood — turned out to be the most influential British
mathematician of the first half of the 20th century. They are credited for
boosting British mathematics to the top level again, after it had stayed
behind the achievements in France and Germany during the 19th century. In
particular, Hardy insisted on mathematical rigor, which was in total contrast
to Ramanujan’s intuitive style [3].

Unlike his colleagues, Hardy became aware of the enormous value of Ramanujan’s
results, although part of it had been known before, and some formulae were
wrong — but the rest was groundbreaking [4]. Having received two letters with
120 remarkable equations, Hardy invited Ramanujan to Cambridge, which he
accepted after some hesitation, and where he stayed from 1914 to 1919, i.e.
mostly during World War I. It was not easy for him to get used to the climate,
lifestyle and food.444Being a devout Hindu, Ramanujan was a strict vegetarian,
which was highly unusual in England at that time. Moreover, he suffered from
serious health problems; they had antecedents in his earlier life in India,
and they lead to his decease one year after his return to Madras.

Despite appreciating his brilliance, Hardy urged him to take lectures (for
instance, Ramanujan hardly knew anything about complex analysis), and in
particular he insisted on proofs, not just conjectures. That was not easily
compatible with Ramanujan’s mentality, but he published 32 high-impact papers
during his 5 years in Cambridge, 7 of them together with Hardy [5]. In 1918
Ramanujan was elected as a Fellow of the Royal Society, as one of the youngest
members in its history, and half a year later he also became a Fellow of the
Cambridge Trinity College.

Figure 2: Godfrey Hardy (1877–1947)

His investigations involved subjects, which had been considered intractable
before, in particular a miraculous approximate formula for partitioning, which
(surprisingly) involves the number $\pi$ [6]. Ramanujan traced this number in
all kind of contexts; best known is a series that he postulated in Ref. [7]
(along with a variety of other $\pi$-approximation formulae),

$\frac{1}{\pi}=\frac{\sqrt{8}}{99^{2}}\ \sum_{n\geq
0}\,\frac{(4n)!}{(4^{n}n!)^{4}}\ \frac{1103+26390n}{99^{4n}}\ .$ (1.1)

It converges exponentially (despite the factor $(4n)!$ in the numerator), thus
it provides one of the fastest algorithms to compute $\pi$. If we truncate at
$n_{\rm max}=0,\,1,\,2$, we obtain the corresponding approximation
$\pi_{n_{\rm max}}$, which differs from the exact value of $\pi$ as

$|\pi-\pi_{0}|\simeq 7.6\cdot 10^{-8}\ ,\quad|\pi-\pi_{1}|\simeq 6.4\cdot
10^{-16}\ ,\quad|\pi-\pi_{2}|\simeq 5.7\cdot 10^{-24}\ .$ (1.2)

How Ramanujan arrived at such formulae is hard to know: Hardy later described
it as a “process of mingled argument, intuition, and induction, of which he
was entirely unable to give any coherent account” [8].

Here we are going to address a relatively simple subject, which Ramanujan
mentioned in his first letter to Hardy [9], and which he had written down
before in Chapter VI of his Second Notebook [10]. This letter contains two
apparently weird formulae for divergent series,

$\displaystyle\sum_{n\geq 1}^{\rm(R)}n=1+2+3+4+5+\dots=-\frac{1}{12}\ ,$ (1.3)
$\displaystyle\sum_{n\geq 1}^{\rm(R)}n^{3}=1+8+27+64+125+\dots=\frac{1}{120}\
,$ (1.4)

where the sums run from $n=1\dots\infty$, and the superscript (R) indicates
“Ramanujan summation” [11]. These strange relations have fascinated
generations of people; for instance a discussion of eq. (1.3) in YouTube [12],
dated 2016, has over 2.4 million views, and over 5000 quite controversial
comments.

Of course, it is provocative to write these relations as straight equations,
as Ramanujan did (without any superscript), but it fulfills the purpose of
attracting attention and causing debate. Still, in the following we are going
to replace the symbol $=$ by $\stackrel{{\scriptstyle\wedge}}{{=}}\,$, meaning
“corresponds to” or “is associated with”. In this sense, we are going to show
that the fractional values on the right-hand side do have a meaning, not only
as a mathematical peculiarity, but they can even be used to derive physical
results.

Unlike other addressees of Ramanujan’s letters, Hardy recognized the values of
the Riemann $\zeta$-function, or $p$-series. For ${\rm Re}\,z>1$ it is defined
as

$\zeta(z)=\sum_{n\geq
1}\frac{1}{n^{z}}=\frac{1}{1^{z}}+\frac{1}{2^{z}}+\frac{1}{3^{z}}+\frac{1}{4^{z}}+\dots\
.$ (1.5)

In 1739 Leonhard Euler had computed explicit expressions for $\zeta(2n)$,
$n\in\mathbb{N}_{+}$, and later he also conjectured a $\zeta$-functional
relation [13]. More than a century later, in 1859, Bernhard Riemann
established the analytic continuation of the $\zeta$-function to
$\mathbb{C}-\\{1\\}$ [14], see Appendix C. In this sense, Hardy noticed that
Ramanujan’s results can be interpreted as $\zeta(-1)$ and $\zeta(-3)$
(although these values are not explicitly given in Ref. [14]).

Riemann was a leading mathematician of the 19th century, and of all times, cf.
footnote 2. Like Ramanujan he lived his youth in harsh poverty, until he was
appointed to a post in Göttingen, on Carl Friedrich Gauss’ recommendation.
Another analogy is that he soon suffered from health problems. Hoping that a
warmer climate might help against his tuberculosis [15] (which was also among
Ramanujan’s diseases [3]), he spent extended periods in Italy, where he died
in 1866, at the age of 39.

Figure 3: Bernhard Riemann (1826–66)

Differences from Ramanujan’s life are that Riemann had access to education at
leading mathematical institutes, in Göttingen and Berlin, and that he
published his results only after elaborating rigorous proofs. His publications
had an enormous impact, but only Ref. [14] deals with number theory. There he
discussed the density of prime numbers,555This is another field of common
interest of these two geniuses: later Ramanujan proposed his own formula for
the prime number density, which is, however, not as accurate as he had
expected. and it was in this context that he postulated the analytic
continuation of the $\zeta$-function; the crucial functional equation is
displayed in eq. (C.1). In contrast to Ramanujan, Riemann was an expert on
complex analysis. Presumably he had hand-written notes with many more
important results, but after his sudden death his house-cleaner burned part of
these notes, until some mathematicians managed to stop her [16].

Ramanujan did not provide an actual derivation of formulae (1.3) and (1.4),
but in the first case he assigned — in Ref. [10] and also his first letter to
Hardy — a value to another divergent series, as an intermediate step to arrive
at relation (1.3). That series corresponds to a special case of Dirichlet’s
$\eta$-function, or alternating $\zeta$-function,

$\eta(z)=\sum_{n\geq
1}\frac{(-1)^{n-1}}{n^{z}}=\frac{1}{1^{z}}-\frac{1}{2^{z}}+\frac{1}{3^{z}}-\frac{1}{4^{z}}\dots\
,$ (1.6)

which converges for ${\rm Re}\,z>0$. At ${\rm Re}\,z>1$ we obtain

$\zeta(z)-\eta(z)=2\sum_{n\geq 1}\frac{1}{(2n)^{z}}=2^{1-z}\zeta(z)\
,\quad\zeta(z)=\frac{1}{1-2^{1-z}}\,\eta(z)\ .$ (1.7)

The latter defines $\zeta(z)$ in the domain ${\rm Re}\,z>0\wedge z\neq 1$.

In particular, Ramanujan wrote down its continuation to [10]

${\cal E}:=\eta(-1)=\sum_{n\geq
1}(-1)^{n-1}n=1-2+3-4\dots\stackrel{{\scriptstyle\wedge}}{{=}}\frac{1}{4}\ .$
(1.8)

We are going to confirm this value, and follow his path to relations (1.3) and
(1.4), which we finally apply to physics, in particular to the Casimir effect.

##  2 Heuristic derivation of $\sum_{n\geq
1}n\stackrel{{\scriptstyle\wedge}}{{=}}-1/12$

Series have both fascinated and confused mathematicians over and over again,
throughout history. The famous “paradox” by Zeno, which describes a race
between Achilles and a tortoise (and further “paradoxes” of a similar style)
caused a deep crisis in the mathematics of Ancient Greece (see e.g. Ref.
[17]), because the concept of convergent series — in this case, a geometrical
series — had not yet been understood.

Here we just take the familiar geometrical series as the point of departure.
For $|z|<1$ we trivially obtain

$G(z)=\sum_{n\geq 0}z^{n}=1+z+z^{2}+z^{3}\dots=1+z\,G(z)\ \ \Rightarrow\ \
G(z)=\frac{1}{1-z}\ .\ $ (2.1)

The series converges only for $|z|<1$, but the final function $G(z)$ is
defined all over $\mathbb{C}-\\{1\\}$. Moreover, the complex function $G(z)$
is holomorphic (or complex analytic, i.e. complex differentiable)666We recall
that this is a powerful property, which guarantees that the function has
derivatives of any order in its domain of holomorphy, and that it coincides
with its power series. Moreover, since $G^{\prime}(z)\neq 0$ it is also
conformal, i.e. angle conserving: if we interpret the function $G(z)$ as a map
$\mathbb{C}\to\mathbb{C}$, and consider two curves $\gamma_{1}(z)$,
$\gamma_{2}(z)$, which intersect in $z_{0}$ with a certain angle, then the
maps of these curves intersect in $G(z_{0})$ with the same angle. in
$\mathbb{C}-\\{1\\}$, and therefore meromorphic in $\mathbb{C}$, which implies
that its analytic continuation from the disk $|z|<1$ to $\mathbb{C}-\\{1\\}$
is unique, cf. Appendix C.

This allows us to define Grandi’s series

${\cal G}=1-1+1-1+1-1\dots=\sum_{n\geq 0}(-1)^{n}$ (2.2)

by means of analytic continuation,

${\cal G}\stackrel{{\scriptstyle\wedge}}{{=}}G(z)|_{z=-1}=\frac{1}{2}\ .$
(2.3)

We can readily extend this scheme to the $\eta$-function in eqs. (1.6), (1.8).
To this end, we first return to safe grounds, i.e. to $|z|<1$, where

$G(z)^{2}=1+2z+3z^{2}+4z^{3}\dots=\sum_{n\geq
1}n\,z^{n-1}=G^{\prime}(z)=\frac{1}{(1-z)^{2}}\ .$ (2.4)

The function $G(z)^{2}$ is holomorphic as well, again with a (unique) analytic
continuation to $\mathbb{C}-\\{1\\}$. This implies in particular

${\cal E}=\sum_{n\geq
1}(-1)^{n-1}n\stackrel{{\scriptstyle\wedge}}{{=}}G(z)^{2}|_{z=-1}={\cal
G}^{2}=\frac{1}{4}\ ,$ (2.5)

which coincides with Ramanujan’s result (1.8).

However, this is not yet what we need in order to assign a value to the
notorious series, which we denote as

${\cal R}:=\sum_{n\geq 1}n=1+2+3+4+5+\dots\ .$ (2.6)

It would correspond to $G(z)^{2}|_{z=1}$, but $z=1$ is just the point where
this function has its double pole. Following Ramanujan’s line of thought [10],
we proceed by introducing another series

$R_{1}(z)=1-2z+3z^{2}-4z^{3}+5z^{4}\dots=\sum_{n\geq 1}n(-z)^{n-1}\ ,$ (2.7)

which also converges at $|z|<1$, and we formally obtain ${\cal
R}\stackrel{{\scriptstyle\wedge}}{{=}}R_{1}(-1)$.777The reason for the
notation with an index 1 will become clear in Appendix B. Again we refer to
the safe region, i.e. to the disk $|z|<1$, where we take the difference

$G(z)^{2}-R_{1}(z)=4z(1+2z^{2}+3z^{4}+\dots)=4z\sum_{n\geq 1}nz^{2(n-1)}\ .$
(2.8)

This operation can only be justified inside the convergence disk, but once it
is carried out, taking the limit $z\to-1$ on both sides leads to

$^{\lim}_{z\to-1}\Big{[}G(z)^{2}-R_{1}(z)\Big{]}=\frac{1}{4}-{\cal
R}\stackrel{{\scriptstyle\wedge}}{{=}}-4{\cal R}\quad\Rightarrow\quad{\cal
R}\stackrel{{\scriptstyle\wedge}}{{=}}-\frac{1}{12}\ .$ (2.9)

Thus Ramanujan removed the divergence in a controlled manner, which leaves an
unambiguous finite value, and Hardy noticed that this assignment corresponds
to ${\cal R}=\zeta(-1)$. In Appendix B we will discuss what has been going on
here.

Alternatively we could invoke eq. (1.7) and consider the divergent series
${\cal C}=1+1+1+1+1\dots$. The limit $z\to 0$ implies

${\cal
C}\stackrel{{\scriptstyle\wedge}}{{=}}\zeta(0)\stackrel{{\scriptstyle\wedge}}{{=}}-\eta(0)\stackrel{{\scriptstyle\wedge}}{{=}}-{\cal
G}=-\frac{1}{2}\ ,$ (2.10)

a value which is also given in Ramanujan’s Second Notebook [10]. When we even
insert $z=-1$ in eq. (1.7), we arrive again at

${\cal R}\stackrel{{\scriptstyle\wedge}}{{=}}-\tfrac{1}{3}{\cal
E}\stackrel{{\scriptstyle\wedge}}{{=}}-\frac{1}{12}=\zeta(-1)\ .$ (2.11)

So far this may look like a mathematical playground, but in the next section
we are going to apply this result to a physical toy model, where it leads to
sensible results. In Section 4 we will proceed to a setting, which refers to
physical phenomenology; for that purpose we will need relation (1.4), which
corresponds to $\zeta(-3)$.

##  3 Casimir effect on a line

In this section and beyond, we are going to deal with quantum field theory.
General introductions can be found in a number of textbooks, such as Refs.
[18] (and a popular science description is given in Ref. [19]), but in order
to follow the derivations in Sections 3 and 4 only very little knowledge about
it is required. Our notation implicitly refers to the functional integral
formulation, where the fields are functions of the space and time variables,
with values which can, for instance, be real numbers (then it is a neutral
scalar field, as in this section), or vectors (as in Section
4).888Alternatively, in the canonical formalism the fields are operator-
valued. In general, all field configurations — i.e. all possible values in
each space-time point — are integrated to obtain expectation values of
observables.

Here, however, we are only concerned with the ground state contributions of
free fields. For a neutral scalar field we can imagine an (infinite) set of
coupled harmonic oscillators, one at each space point. A Fourier transform
yields oscillators for all possible frequencies. A priori these frequencies
are not restricted, so if we sum up their ground state contributions to the
vacuum energy density, the result diverges.

We are going to be confronted with these ultraviolet (UV) divergences: they
require a regularization, i.e. a mathematical modification which makes such
sums (or integrals) finite, enabling calculations. In the end we want to
remove the regularization, hence we aim at a cancelation of the UV
divergences. This can often be achieved by subtracting divergent terms, so-
called counterterms, which correspond to some limit; without taking that
limit, a finite quantity remains. This procedure is known as a
renormalization: it should lead to finite results for the physical quantities,
which do not depend on the regularization that has been chosen (if suitable
conditions are fulfilled).

The concepts, which we have sketched here in an abstract form, are going to be
illuminated by the presentation of simple examples.

To this end, we address an effect, which was theoretically predicted by the
Dutch physicists Hendrik Casimir and Dirk Polder in 1947/8 [20]. In
particular, we follow the perspective that Casimir adopted a little later
[21], inspired by a discussion with Niels Bohr.

Figure 4: Hendrik Casimir (1909–2000)

As a toy model, we consider a free, neutral, massless scalar field on a line,
$\phi(t,x)\in\mathbb{R}$, $x\in\mathbb{R}$. At the points $x=0$ and $x=d>0$ we
apply Dirichlet boundaries, which force the field to vanish, i.e.
$\phi(t,0)=\phi(t,d)=0$. In this interval the field configurations can be
Fourier decomposed into standing waves with wavenumbers $k_{n}=n\pi/d$,
$n=1,2,3\dots$ (such that $\sin(k_{n}d)=0$). In natural units ($\hbar=c=1$),
they contribute $k_{n}/2$ to the ground state energy.

For the vacuum energy density in this interval, we formally obtain

$\rho_{d}=\frac{1}{2d}\sum_{n\geq 1}k_{n}=\frac{\pi}{2d^{2}}\sum_{n\geq 1}n\
,$ (3.1)

where we encounter the divergent term ${\cal R}$, to which Ramanujan assigned
the value $-1/12$. We are now going to illustrate — in a physical framework —
why, and in which sense, this value is indeed meaningful.

As usual in quantum field theory, we first need a regularization (as we
mentioned before), but it doesn’t need to be fully specified. We regularize
$\rho_{d}$ by performing a substitution

$n\to f(n)=n\,r(n/\Lambda d)\ ,\quad{\rm with}\quad r(0)=1\ ,$ (3.2)

where $f$ and $r$ are smooth functions on $\mathbb{R}_{0}^{+}$ (an infinite
number of times continuously differentiable), and $\Lambda$ is an energy
cutoff. If we remove it, $\Lambda\to\infty$, we recover the term before
regularization. At finite $\Lambda$ we require

$^{\lim}_{x\to+\infty}\,f(x)=0\ ,\quad^{\lim}_{x\to+\infty}\,f^{(k)}(x)=0\ ,$
(3.3)

where $f^{(k)}$ is any odd derivative $(k=1,\,3,\,5,\dots).$

A simple example of $f(n)$ in eq. (3.2) is the heat kernel regularization,
where the function $r$ is exponential, $f(n)=n\exp(-n/\Lambda d)$, which leads
to a geometrical series,

$\displaystyle\sum_{n\geq 1}ne^{-n/d\Lambda}$ $\displaystyle=$
$\displaystyle\Lambda^{2}d\ \frac{\partial}{\partial\Lambda}\sum_{n\geq
0}e^{-n/\Lambda d}=\Lambda^{2}d\
\frac{\partial}{\partial\Lambda}\frac{1}{1-\exp(-1/\Lambda d)}$ (3.4)
$\displaystyle=$ $\displaystyle(\Lambda d)^{2}-\frac{1}{12}+{\cal
O}\Big{(}\frac{1}{\Lambda d}\Big{)}\ .$

We already see ${\cal R}=-1/12$ popping up, but we want to proceed to a
broader perspective, which generalizes the regularization, and which also
clarifies the rôle of the UV divergent term, along the lines of Ref. [21].

In an infinite interval, $d\to\infty$, the formal term (3.1) for the energy
density turns into a momentum integral. We expand the difference
$\rho_{d}-\rho_{\infty}$, at the regularized level, by means of the Euler-
Maclaurin formula,

$\sum_{n=1}^{N}f(n)-\int_{0}^{N}f(x)\,dx=\frac{f(N)-f(0)}{2}+\sum_{j\geq
1}\frac{B_{2j}}{(2j)!}\Big{(}f^{(2j-1)}(N)-f^{(2j-1)}(0)\Big{)}\ ,$ (3.5)

where $N\in\mathbb{N}_{+}$. A finite number $N$ represents another
component of the UV regularization: in this case, we sum over $k_{n}$ only up
to $k_{n,{\rm max}}=N\pi/d$.

The powerful formula (3.5) was independently derived by Euler and by Colin
Maclaurin around 1735. It is very useful in field theory, in particular when
dealing with finite temperature or finite-size effects. Since we assume the
function $f$ to be smooth and to fulfill the condition (3.3), this series
converges both at finite $N$ and in the limit $N\to\infty$.999If we truncated
this expansion at some odd integer $J/2$, such that we deal with
$\sum_{j=1}^{J/2}\dots$, then there is a remainder term $R_{J}$ on the right-
hand side. It can be estimated as $|R_{J}|\leq 2\zeta(J)\int_{1}^{N}dx\
|f^{(J)}(x)|/(2\pi)^{J}$ [22], hence the above assumptions imply
${}^{\lim}_{J\to+\infty}R_{J}=0$. The coefficients in the last term are the
Bernoulli numbers,101010Bernoulli numbers were a particular passion of
Ramanujan, who had certainly read about them in Ref. [2]. His very first paper
discussed their properties [23]. For instance, he showed that the denominators
of $B_{2}$, $B_{4}$, $B_{6}$, $B_{8}$ $\dots$ (in lowest terms) all contain
the prime factors 2 and 3 exactly once. which can be defined in a way related
to eq. (3.4),

$\frac{x}{1-e^{-x}}=\sum_{k\geq 0}B_{k}\frac{x^{k}}{k!}\ .$ (3.6)

This yields $B_{0}=1$, $B_{1}=\frac{1}{2}$, $B_{2}=\frac{1}{6}$,
$B_{4}=-\frac{1}{30}$, $B_{3}=B_{5}=B_{7}=\dots=0$,111111This becomes obvious
if we define $g(x)=x/(1-e^{-x})$ and compute $g(x)-g(-x)=x$. ($B_{6}$,
$B_{8}\dots$ do not vanish, but we won’t need them).

We insert in eq. (3.5) a function $f$ which fulfills the conditions (3.2) and
(3.3), and we take the UV limit in two steps: first we let $N\to\infty$; due
to eq. (3.3) all contributions vanish in this limit. As for the terms at
$x=0$, we note that $f(0)=0$, $f^{\prime}(0)=r(0)=1$, $f^{(k)}(0)={\cal
O}((\Lambda d)^{1-k})$, $k=2,\,3,\dots$, hence the second step of the UV
limit, $\Lambda\to\infty$, leads to

$\rho_{d}-\rho_{\infty}=\frac{\pi}{2d^{2}}\Big{(}-\frac{1}{2}B_{2}\Big{)}=-\frac{\pi}{24d^{2}}\
,$ (3.7)

where ${\cal R}=-B_{2}/2=-1/12$ is crucial, in agreement with eq. (3.4). We
recover Ramanujan’s assignment of a finite value to the divergent series in
eq. (3.1), i.e. Ramanujan summation corresponds to the subtraction of the
infinite-volume limit of the vacuum energy density. This density diverges both
in a finite and in an infinite interval, but the difference, i.e. its finite-
size effect, is finite and well-defined. The elimination, or isolation, of a
UV divergent term ($(\Lambda d)^{2}$ in eq. (3.4)), in order to deal with
finite differences, is the basic idea of renormalization. In field-theoretic
jargon, we have subtracted the counterterm $\rho_{\infty}$, which cancels the
divergence in the series (3.1).

Interestingly, this recipe matches exactly the relation

$\zeta(-1)=\sum_{n\geq 1}^{\rm(R)}n=-\frac{1}{2}B_{2}=-\frac{1}{12}\ ,$ (3.8)

applied to eq. (3.1). We recall that the superscript (R) means Ramanujan
summation; its general properties are defined and explored in Ref. [11]. For
the purpose of this article, it is sufficient to point out that for a series
of the form $\sum_{n\geq 1}^{\rm(R)}n^{k}$, $k\in\mathbb{N}$, the Ramanujan
summation coincides with the $\zeta$-function $\zeta(-k)$ (defined by analytic
continuation), see Appendix C. It further corresponds to the finite term in
the Euler-Maclaurin expansion of the difference
${}^{\lim}_{N\to\infty}[\sum_{n=1}^{N}n^{k}-\int_{0}^{N}x^{k}\,dx]$, which can
be read off from eq. (3.5), and which generalizes eq. (3.8) to

$\zeta(-k)=\sum_{n\geq 1}^{\rm(R)}n^{k}=-\frac{B_{k+1}}{k+1}\ ,\quad
k\in\mathbb{N}\ .$ (3.9)

The question remains: beyond the satisfaction of deducing a finite result, why
are we interested in this difference? What is its physical meaning? One is
tempted to reply: the change of the vacuum energy, as a function of $d$,
implies a force between the Dirichlet boundaries, and the counterterm can be
subtracted since it does not depend on $d$, so it does not contribute to this
force. However, there is still a caveat, which is often ignored: the
boundaries could also affect the energy outside the interval $[0,d\,]$. That
could contribute to the force between the boundaries, so we have to be
careful.

A sound approach introduces three Dirichlet boundaries, at the points $0$,
$d$, $L$, with $0<d<L$, see Figure 5 (left).

Figure 5: Left: Setting for the Casimir effect on a line: we apply Dirichlet
boundaries at the positions $0$, $d$ and $L$, with $0<d<L$. Right: The Casimir
force $F(d)$, which acts on the “piston” at $d$ according to eq. (3.11), in
units such that $L=1$.

The idea is to keep the extreme boundaries at $0$ and $L$ fixed, while the one
at $d$ is a variable “piston”. In this way, the energy outside the interval
$[0,L]$ remains constant, while the energy inside this interval can be
computed explicitly, so everything is under control. From eq. (3.7) we obtain
the total vacuum energy

$E(d)=-\frac{\pi}{24d}-\frac{\pi}{24(L-d)}+E_{\rm out}=-\frac{\pi
L}{24d(L-d)}+E_{\rm out}\ .$ (3.10)

The term $E_{\rm out}$, which represents the energy outside the interval
$[0,L\,]$, is divergent, but it does not depend on $d$. Generally a force is
obtained as the negative gradient of the potential energy. In our
1-dimensional case, this operator reduces to the negative derivative with
respect to $d$ (the only variable involved). Therefore the term $E_{\rm out}$
is irrelevant for the force acting on the “piston” at $d$, which is obtained
as

$F(d)=-E^{\prime}(d)=-\frac{\pi L}{24}\frac{L-2d}{d^{2}(L-d)^{2}}\ ,$ (3.11)

and depicted in Figure 5 (right). It is odd with respect to the center
$d=L/2$, and attractive towards the nearer fixed boundary, at $0$ or $L$.
Hence $d=L/2$ is an unstable equilibrium position. In the case $L\gg d$ we
obtain a force, which is attractive towards the boundary at $0$,
$F(d)\simeq-\pi/(24d^{2})$, and which coincides with the 2-boundary picture of
eq. (3.7). Hence, in that picture, ignoring effects outside the interval
$[0,d\,]$ is justified after all (varying $d$ does not change the energy in
the half-line with $x>d$).

To summarize this section, we have renormalized the system by discarding an
additive, infinite constant in the energy density, the counterterm
$\rho_{\infty}$, which represents the infinite-volume limit. In order to
compute the remaining finite term — a finite-size effect in this case — a
regularization is needed. Then the Euler-Maclaurin formula can be applied, and
by removing both UV cutoffs, $k_{n,{\rm max}}=Nd/\pi\to\infty$ and
$\Lambda\to\infty$, we arrive at the finite result (3.7). It does not depend
on the choice of the regularizing function $f$, as long as the conditions
(3.2) and (3.3) are fulfilled. This leads to finite values for the vacuum
energy in the interval $[0,L]$, and for the force $F(d)$ in eq. (3.11), which
acts on the “piston”.

##  4 The Casimir force in 3-dimensional space

We proceed to a realistic situation, which deals with the vacuum energy of the
photon field in $(3+1)$-dimensional space-time. The simplest setting is shown
Figure 6: it involves two parallel, conducting plates,121212In theory, we
assume perfect conductivity, this is what it takes to implement exact
Dirichlet boundaries. The experiments have been performed with well-conducting
metal plates, which provide a good approximation, cf. Section 5.2. The
generalization with respect to the dielectric constant was theoretically
studied by Evgeny Lifshitz [24]. of the same rectangular shape and area $A$,
separated by a short distance $d$.131313Throughout this article, we refer to
the standard scenario with static Dirichlet boundaries. The two-fold
generalization of the Casimir effect with dynamical Robin boundaries is
discussed for scalar fields in Refs. [25].

Figure 6: Setting for an experimental test of the Casimir effect: one measures
the force between two parallel, conducting plates of area $A$, separated by a
short distance $d$.

We rely on our experience from the 1-dimensional toy model to conjecture that
it is sufficient to consider the energy $E(d)$ between the plates. This is
appropriate when the area $A$ is large ($\sqrt{A}\gg d$). The photon momentum
components parallel to the plates — which we denote as $k_{1}$, $k_{2}$ — are
treated as continuous. Hence we perform a discrete sum, in analogy to eq.
(3.1), only over the vertical component $k_{3}$, and the energy between the
plates takes the form

$\displaystyle E(d)=A\,d\,\rho(d)=\frac{1}{2}A\frac{1}{(2\pi)^{2}}\int
dk_{1}dk_{2}\ \sum_{n\geq 0}2\sqrt{k_{1}^{2}+k_{2}^{2}+\Big{(}\frac{\pi
n}{d}\Big{)}^{2}}$ $\displaystyle=\frac{A}{2\pi}\sum_{n\geq
0}\int_{0}^{\infty}dK\ K\sqrt{K^{2}+\Big{(}\frac{\pi
n}{d}\Big{)}^{2}}=\frac{A}{6\pi}\sum_{n\geq
0}\left.\Big{[}K^{2}+\Big{(}\frac{\pi
n}{d}\Big{)}^{2}\Big{]}^{3/2}\right|_{0}^{\infty}\ ,\qquad$ (4.1)

where we have inserted a factor $2$ for the two photon polarization states.

Note that we have not regularized so far. If we do so and follow the procedure
of Section 3, we can renormalize by subtracting the energy in the same volume
but without plates, $E_{\infty}=A\,d\,\rho_{\infty}$.141414To obtain
$E_{\infty}$ we start from the term in the upper line of eq. (4.1), convert
(in the large-$d$ limit) $\pi n/d$ to the continuous momentum component
$k_{3}$, and $\sum_{n\geq 0}$ to $(d/\pi)\int_{0}^{\infty}dk_{3}$, which leads
to $E_{\infty}=A\,d\,(2\pi)^{-3}\int d^{3}k\,|\vec{k}|$. In this difference,
first the UV contribution due to $K\to\infty$ cancels (a physical
interpretation is that infinitesimally short wavelengths are not sensitive to
the presence of boundaries at a finite distance). Regarding $K=0$, we apply
the Euler-Maclaurin expansion (3.5) to $\sum_{n\geq
1}\dots-\int_{0}^{\infty}dk_{3}\dots$. This corresponds to the Ramanujan
summation over $n$, which we again express as a $\zeta$-function,

$\displaystyle\frac{E(d)}{A}$
$\displaystyle\stackrel{{\scriptstyle\wedge}}{{=}}$
$\displaystyle-\frac{1}{6\pi}\frac{\pi^{3}}{d^{3}}\zeta(-3)=-\frac{\pi^{2}}{720d^{3}}\
,$ $\displaystyle\frac{F(d)}{A}$ $\displaystyle=$
$\displaystyle\frac{-E^{\prime}(d)}{A}=-\frac{\pi^{2}}{240d^{4}}\ .$ (4.2)

We have used eq. (3.9) and inserted $B_{4}=-1/30$, in agreement with eq.
(1.4). Although the sign of $B_{4}$ is opposite to $B_{2}$ (which we inserted
in eq. (3.7)), we obtain again an attractive force between the Dirichlet
boundaries: note that there is another sign flip due to the integral over $K$,
where the lower bound contributes.

The question arises of what magnitude this force takes for realistic sizes $A$
and $d$, and if such a force can be measured. The first conclusive experiment
was achieved in 1997 by Steve Lamoreaux, who succeeded in measuring the
Casimir force to $5\,\%$ accuracy [26]. This was soon followed by Umar
Mohideen and Anushree Roy [27], who took into account the corrections due to
finite temperature, finite conductivity and the roughness of the surfaces. In
these experiments, the geometrical structure was a plate and a sphere, because
of the difficulty in keeping two plates parallel to very high precision.

The first experiment to successfully measure the Casimir force between
parallel plates, as sketched in Figure 6, was carried out at the University of
Padua, Italy, in 2002 [28], with $15\,\%$ precision. That experiment used
rectangular silicon stripes, covered with a chromium layer, of size
$A=1.9\,{\rm cm}\times 1.2\,{\rm mm}$, and their separation $d$ varied from
$0.5\,\mu{\rm m}$ to $3\,\mu{\rm m}$. In order to compute the predicted force
in Newton (N), we have to insert a factor $\hbar c$ in eq. (4.2),151515We
recall that we have been using natural units. This factor shows that we are
dealing with a relativistic quantum effect. which leads to

$F(d)\simeq-1.3\cdot 10^{-7}\,{\rm N}\,\Big{(}\frac{\mu{\rm
m}}{d}\Big{)}^{4}\,\frac{A}{{\rm cm}^{2}}\ .$ (4.3)

Hence the predicted force in this experiment varied between $F\simeq-4.7\cdot
10^{-7}~{}{\rm N}$ and $-3.7\cdot 10^{-10}~{}{\rm N}$. Forces of this range
are in fact measurable: for instance Ref. [26] used a torsion pendulum and
laser interferometry, and Ref. [27] employed an atomic force microscope. In
the experiment reported in Ref. [28], one of the parallel plates was the face
of a cantilever beam, free to oscillate. The variation of the force was
observed by measuring shifts in its resonator frequency, employing a fiber-
optic interferometer.

The precision in recent experiments is around, or below, $1\,\%$.

##  5 Concluding remarks

We don’t know what exactly Ramanujan had in mind when he introduced his
summation of divergent series, which we now denote as Ramanujan summation,
such as relations (1.3) and (1.4). In his letter to Hardy he only documented
one intermediate step, relation (2.5). In his Second Notebook [10] he
additionally hinted at the continuation (2.9), and he mentioned the difference
between summation and integration, which we also reviewed. This is a valid
argument, and apparently Ramanujan reinvented an equivalent form of the Euler-
Maclaurin formula (he did not use that term [10], nor does this formula appear
in Ref. [2]).

A reason for the sparse documentation in Ramanujan’s notebooks was — in
addition to his intuitive way of thinking — that he mostly worked on a slate
and only wrote down final results on paper, which was valuable (in particular
for him, who was living in poverty). It is also conceivable that he was
influenced by Carr’s telegram style [2], cf. footnote 3. In any case, his
approach matches the analytic continuation of the $\zeta$-function, at least
with respect to negative integer arguments, see eq. (3.9). In fact, he also
rediscovered the analytically continued $\Gamma$-function with values in
$\mathbb{C}-\\{0,-1,-2\dots\\}$. He highlighted this idea in the introduction
of his first letter to Hardy [9], unaware that this had been known before; in
particular, Riemann had used it in Ref. [14].

These finite values for divergent series may look like a mathematical game,
which is rather disconnected from reality. However, it is possible to
establish consistent rules for the Ramanujan summation of divergent series by
carefully dealing with properties like linearity and translation [11].
Moreover, we reviewed their striking application to physics, where they enable
the prediction of a force, which has in fact been measured. The meaning of the
Casimir effect will be discussed in the following two subsections.

###  5.1 Is the electromagnetic vacuum energy density real?

Numerous authors infer from the experimental observation of the Casimir force
the existence of the vacuum energy of the photon field, $\rho_{\rm vac}$, as
predicted by Quantum Electrodynamics (QED), e.g. Refs. [26, 27, 28, 29, 30,
31]. For instance, Ref. [31] states that “the existence of zero-point vacuum
fluctuations has been spectacularly demonstrated by the Casimir effect.” It
does, however, not affect usual experiments, which only depend on energy
differences, not on the additive constant $\rho_{\rm vac}$. Still, such an
energy density throughout the Universe, known as Dark Energy, is indeed
manifest since it affects the expansion of the Universe.

It corresponds to the Cosmological Constant in General Relativity: in its
absence — which was generally assumed from the 1930s to the 1990s — the
expansion of the Universe would be decelerated. However, at the very end of
the 20th century it was observed that the expansion is accelerated.161616This
was concluded from the distance and redshift of a set of type Ia supernovae
[32]. This is best described by a positive Cosmological Constant, which
corresponds to a Dark Energy density of about $\rho_{\rm
DE}\approx(0.002~{}{\rm eV})^{4}$.

Unfortunately this value is totally incompatible with the vacuum energy
density $\rho_{\infty}$ that we discussed. First, $\rho_{\infty}$ seems to
diverge, as we saw, but one might impose an UV cutoff in the integral $\int
d^{3}k\ |\vec{k}|$, most naturally at the Planck energy. This leads to a
finite value $\rho_{\rm Planck}$, which is, however, much too large,
$\rho_{\rm Planck}/\rho_{\rm DE}={\cal O}(10^{121})$.

People who still believe in supersymmetry could argue that in a perfectly
supersymmetric world the Dark Energy vanishes (since bosons and fermions
appear in pairs of the same mass, and the fermionic ground state energy is
negative, with the same absolute value [18]). However, even if supersymmetry
exists, it has to be badly broken in our low-energy world (otherwise particles
like the “selectron” would have been observed), and the required extent of
breaking still implies a Dark Energy density, which exceeds $\rho_{\rm DE}$ at
least by a factor ${\cal O}(10^{60})$ [30]. Hence any evidence for the
existence of the QED photon field vacuum energy $\rho_{\rm vac}$ would be
puzzling.

Julian Schwinger et al. computed the Casimir force by means of a source field
technique, without any need to refer to $\rho_{\rm vac}$ [33]. Part of the
literature concludes from that work that the Casimir experiments do not
necessarily imply the reality of $\rho_{\rm vac}$, which could be welcome as a
remedy against the disastrous discrepancy by 121 order of magnitude.

Still, the question remains whether or not relativistic quantum physics could
be formulated without $\rho_{\rm vac}$. If $\rho_{\rm vac}$ exists, in the
field-theoretic sense, one might wonder whether the frequency of a photon is
affected when it passes through regions of different $\rho_{\rm vac}$, e.g.
when it transversally passes through a Casimir cavity, similar to Bernoulli’s
Principle in fluid dynamics. Regarding its vertical motion, there is even a
prediction that the speed of light could be affected [34].

If we wanted to construct a cavity between two conducting plates with
$\rho_{\rm DM}=|\rho_{d}|=\pi^{2}/(720d^{4})$, we would need a separation of
$d\approx 0.3\,\mu{\rm m}$, which happens to be close to the minimal
separation in the Padua experiment.

###  5.2 The nature of the Casimir force

One could question what kind of force this really is. It does not seem to
appear in the famous list of four forces, which can be described by gauge
fields, nor does it match further interactions in the Standard Model of
particle physics (Yukawa couplings and the Higgs field self-coupling).
However, being an effect of the photon field, this force must ultimately be
electromagnetic, although this is not explicit in the above discussion. From
this perspective, it can be best described as a van der Waals force171717We
refer to the van der Waals force in the narrow sense, also known as London–van
der Waals force, i.e. an induced, attractive, collective multipole interaction
between (electrically neutral) atoms or molecules [35]. between the metal
plates. This is the picture that Casimir and Polder originally had in mind
[20].

So, do we have two equivalent descriptions? This seems puzzling again: in the
van der Waals picture, the force depends on the value of the electromagnetic
coupling constant $\alpha=e^{2}/4\pi\simeq 1/137$, which does not appear
anywhere in the discussion based on the vacuum energy.

This point was analyzed in depth by Robert Jaffe and collaborators, see Ref.
[36] for a summary. They conclude that the Casimir effect is a relativistic
quantum force between electric charges and currents, i.e. a retarded van der
Waals force, which does not require $\rho_{\rm vac}$. In Jaffe’s own words,
“Casimir effects can be formulated and Casimir forces can be computed without
reference to zero point energies.” Thus they contradict the paradigm in this
field, but this issue remains controversial.

Ref. [36] obtains a Casimir force, which does depend on $\alpha$, such that
$F(\alpha=0)=0$, and $F(\alpha\simeq 1/137)$ is the physical strength. In this
exceptional case, even the limit $\alpha\to\infty$ leads to a finite result:
$F(\alpha\to\infty)$ just matches the force obtained from $\rho_{\rm vac}$,
which is often close to $F(\alpha\simeq 1/137)$. For instance, for copper
plates separated by $0.5\ \mu{\rm m}$ (which is experimentally realistic), the
consideration with $\rho_{\rm vac}$ is a good approximation if $\alpha\gg
10^{-5}$ [36], which is easily accomplished by the phenomenological value.

A wide-spread objection against that point of view refers to examples, where
the consideration based on $\rho_{\rm vac}$ leads to a repulsive Casimir force
[37], e.g. for specific parallelepipeds [38]. That feature is not easily
encompassed by van der Waals forces.181818On the other hand, Ref. [35]
predicted a repulsive Casimir–van der Waals–type force, which agrees with an
experiment with interacting materials immersed in a fluid [39]. For instance
Lamoreaux [26] writes: “the Casimir and van der Waals forces are quite
different; the van der Waals force is always attractive, whereas the sign of
the Casimir force is geometry dependent.” Ref. [40] disagrees and assigns the
repulsive result to the negligence of cutoff effects. In fact, an approach by
Ricardo Cavalcanti [41], which is manifestly free of any cutoff dependence,
only obtains attractive Casimir forces.

Jaffe and his collaborators insist that the physical Casimir force is always
attractive [36, 40], and therefore compatible with the van der Waals picture.
If this alternative to the paradigm — as expressed in Refs. [26, 27, 28, 29,
30, 31] — is correct, then the approach that we reviewed is not the most
precise one, but it is still in agreement with the experimental results.

Thus, returning to the title of Subsection 5.1, doubts persist about the
physical reality of $\rho_{\rm vac}$ as encoded in QED. There is a consensus,
however, that we do not know how to theoretically derive the Dark Matter
density $\rho_{\rm DE}$, and that we do not understand the enormous
discrepancy from the vacuum energy predicted by quantum field theory.

So far our discussion in this subsection focused on the Casimir effect due to
QED, which was described in Section 4. In principle, such an effect also
exists for other gauge fields, but only for QED it is simple and instructive,
in particular because the photon field does not self-interact. This is also
the only case where the Casimir force is experimentally confirmed.

For instance in Quantum Chromodynamics (QCD) — the gauge theory of the strong
interaction — this effect is much less transparent because of the complicated
self-interaction of the gluon field [18], which occurs since the QCD gauge
group SU(3) is non-Abelian.191919For interacting quantum field theories,
renormalization involves more than subtracting a divergent term, which was
sufficient in Section 4 for the free electromagnetic field. In the interacting
case, one assigns renormalized values to the fields and their couplings, which
are in general energy-dependent. At low energy its behavior is dominated by
non-perturbative effects, which are hard to compute, and which induce an
intricate vacuum structure. Studies in Euclidean space often focus on the rôle
of instantons [42]. For a static quark–anti-quark pair (which is an
idealization), a multipole expansion has been applied to estimate the Casimir
force [43]. Another study [44] deals with the (restricted) Gribov–Zwanziger
action.

Furthermore, there are numerous attempts to theoretically investigate the
gravitational Casimir force, although this is a quantum effect and we do not
have any (fully satisfactory) theory of quantum gravity. Numerous papers refer
to unusual gravitation theories; studies which are (roughly speaking) close to
the framework of General Relativity include Refs. [45]. The question whether
an experimental demonstration of such an effect, with gravitational wave
mirrors, would prove the existence of gravitons is discussed (and negatively
answered) in Ref. [46].

###  5.3 Further physical applications of Ramanujan summation

There are further applications of Ramanujan summation in the perturbative
expansions of quantum field theory, which can be treated by the
$\zeta$-function regularization [47]. It is an alternative to dimensional
regularization, which is most popular in perturbation theory. The
$\zeta$-function regularization removes from the beginning the UV divergent
terms in the Laurent series by inserting $\zeta(-k)$, thus preventing the
necessity of counterterms.202020Its mathematical equivalence to the heat
kernel regularization, see Section 3, was demonstrated by Hardy and Littlewood
[48]. Stephen Hawking advocated its application in curved space-time [49].

Applications of the $\zeta$-function in bosonic string theory are reviewed in
Refs. [50]. Ref. [51] summarizes a key point as follows: a particle mass $m$
is obtained as

$m^{2}=\frac{1}{\sigma}\Big{[}j+\frac{D-2}{2}\sum_{k\geq 1}^{\rm(R)}k\Big{]}\
,$ (5.1)

where $\sigma$ is the string tension, $D$ is the space-time dimension (such
that a worldsheet lives in $D-2$ dimensions), and $j$ is the string excitation
number (here also the Planck scale is set to 1). The term with the sum over
the modes $k$ represents the ground state energy $E_{0}$, where one applies
Ramanujan summation, $E_{0}=-(D-2)/24\sigma$. The case $j=1$ describes spin-1
particles with only two polarization states, which must therefore be massless.
This condition yields the space-time dimension $D=26$, where bosonic string
theory is formulated [50]. The deeper reason is the requirement to cancel the
conformal anomaly. A detailed pedagogical description is given in Ref. [52].

Acknowledgement:  I would like to thank Kimball Milton for instructive
comments. This work was supported by UNAM-DGAPA-PAPIIT, grant number IG100219.

##  Appendix A The failure of partial sums

Many controversial discussions about relations like (1.3) — for instance
numerous comments on Ref. [12] — refer to partial sums of a few summands. In
the framework of divergent series, separating them is conceptually wrong and
leads to contradictions. It is entertaining to look at some examples, to see
what one should beware of, e.g.

${\cal
R}=1+(2+3+4)+(5+6+7)+(8+9+10)+\dots\mathop{\stackrel{{\scriptstyle?}}{{=}}}1+9{\cal
R}~{}\to~{}{\cal R}\mathop{\stackrel{{\scriptstyle?}}{{=}}}-1/8\ ,\ $ (A.1)

which deviates from Ramanujan’s value. One might even feel tempted to pay
attention to this alternative value, since it is consistent with blocks of any
odd number $u\geq 3$ of summands,

$\displaystyle{\cal
R}=1+2+\dots+\frac{u-1}{2}+\Big{(}\frac{u+1}{2}+\dots+\frac{3u-1}{2}\Big{)}+\Big{(}\frac{3u+1}{2}+\dots+\frac{5u-1}{2}\Big{)}$
$\displaystyle+\dots\mathop{\stackrel{{\scriptstyle?}}{{=}}}\frac{u^{2}-1}{8}+{\cal
R}u^{2}\quad\to\quad{\cal
R}\mathop{\stackrel{{\scriptstyle?}}{{=}}}-\frac{1}{8}\ .$ (A.2)

However, we can show that this approach is even intrinsically inconsistent by
choosing blocks of an even number $g$ summands (where the boundary terms are
equally divided between the blocks),

$\displaystyle{\cal R}$ $\displaystyle=$ $\displaystyle
1+2\dots(\frac{g}{2}-1)+\frac{g}{4}+\Big{(}\frac{g}{4}+(\frac{g}{2}+1)\dots+(\frac{3g}{2}-1)+\frac{3g}{4}\Big{)}$
(A.3) $\displaystyle+$
$\displaystyle\\!\\!\Big{(}\frac{3g}{4}+(\frac{3g}{2}+1)\dots\Big{)}+\dots\mathop{\stackrel{{\scriptstyle?}}{{=}}}\frac{g^{2}}{8}+{\cal
R}g^{2}\quad\to\quad{\cal
R}\mathop{\stackrel{{\scriptstyle?}}{{=}}}-\frac{1}{8}\frac{g^{2}}{g^{2}-1}\
,\qquad\quad$

which only coincides with the claim (A.2) in the limit $g\to\infty$.

Of course, the applicability of partial sums can be disproved more easily,
e.g. in Grandi’s series or Dirichlet’s $\eta$-function, if we write them as

$\displaystyle{\cal G}$ $\displaystyle=$
$\displaystyle(1-1)+(1-1)+(1-1)+\dots=1+(-1+1)+(-1+1)+\dots$
$\displaystyle{\cal E}$ $\displaystyle=$
$\displaystyle(1-2)+(3-4)+(5-6)+\dots=1+(-2+3)+(-4+5)+\dots\quad$

which seems to suggest the contradictory values ${\cal
G}\mathop{\stackrel{{\scriptstyle?}}{{=}}}0$ or ${\cal
G}\mathop{\stackrel{{\scriptstyle?}}{{=}}}1$, and ${\cal
E}\mathop{\stackrel{{\scriptstyle?}}{{=}}}\mp(1+1+1+1\dots)\doteq\mp{\cal C}$.
Again we encounter the series ${\cal C}$, which also appeared in Ramanujan’s
Second Notebook [10], as we anticipated in eq. (2.10). We will come back to it
in Appendix B. In that case, a division into blocks of $n$ summands seems to
suggest ${\cal
C}=(1+\dots+1)+(1+\dots+1)+\dots\mathop{\stackrel{{\scriptstyle?}}{{=}}}n{\cal
C}$, ${\cal C}\mathop{\stackrel{{\scriptstyle?}}{{=}}}0$ or $\infty$ (while
separating $k$ summands ${\cal C}=k+{\cal C},\ {\cal C}=\infty$).

##  Appendix B Analytic continuation from the unit disk

We now follow the scheme of Section 2 by writing the series under
consideration in terms of a variable $z\in\mathbb{C}$, such that they converge
for $|z|<1$, and the divergent series of interest corresponds to the limit
$z\to-1$. Working in the convergence region $|z|<1$ avoids, for instance, the
confusion with partial sums. First, we repeat the geometrical series and
Dirichlet’s $\eta$-function,

$\displaystyle G(z)\\!\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!1+z+z^{2}+z^{3}+\dots=\frac{1}{1-z}\
\stackrel{{\scriptstyle z\to-1}}{{\longrightarrow}}\ {\cal
G}=1-1+1-1\dots\stackrel{{\scriptstyle\wedge}}{{=}}\frac{1}{2}\ .$
$\displaystyle\eta(z)\\!\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\\!1+2z+3z^{2}+4z^{3}+\dots=G(z)^{2}=G^{\prime}(z)=\frac{1}{(1-z)^{2}}$
(B.1) $\displaystyle\stackrel{{\scriptstyle z\to-1}}{{\longrightarrow}}$
$\displaystyle{\cal
E}=1-2+3-4\dots\stackrel{{\scriptstyle\wedge}}{{=}}\frac{1}{4}\ .$

As long as this limit is finite, taking the analytic continuation is trivial.
This allows us to perform operations, which remain valid in the limit
$z\to-1$, like

$\displaystyle\eta(z)+G(z)$ $\displaystyle=$
$\displaystyle\frac{1}{z}[\eta(z)-1]=2+3z+4z^{2}+\dots\quad\stackrel{{\scriptstyle
z\to-1}}{{\longrightarrow}}\ \frac{3}{4}$ $\displaystyle\eta(z)-G(z)$
$\displaystyle=$ $\displaystyle
z\eta(z)=z+2z^{2}+3z^{3}+\dots\quad\stackrel{{\scriptstyle
z\to-1}}{{\longrightarrow}}\ -\frac{1}{4}\ .$ (B.2)

In this framework, the limit $z\to-1$ is well controlled.

This is not obvious anymore when we deal with the pole of the geometrical
series at $z=1$, in particular when we refer to the series

${\cal C}=1+1+1+1+1\dots$ (B.3)

which worried us before in Section 2 and Appendix A. Here we consider three
different regularizing functions (at $|z|<1$),

$\displaystyle C_{1}(z)$ $\displaystyle=$ $\displaystyle
1-z+z^{2}-z^{3}+z^{4}\dots=\frac{1}{1+z}\ ,$ $\displaystyle C_{2}(z)$
$\displaystyle=$ $\displaystyle 1+z^{2}+z^{4}+z^{6}\dots=\frac{1}{1-z^{2}}\ ,$
$\displaystyle C_{3}(z)$ $\displaystyle=$
$\displaystyle-(z+z^{3}+z^{5}+\dots)=-\frac{z}{1-z^{2}}\ .$ (B.4)

We can involve the functions $C_{i}(z)$ in a variety of relations, such as
$G(z)=(1+z)C_{2}(z)=1-(1+z)C_{3}(z)$. Of course they work both in the form of
series and of functions, in agreement with $C_{i}(z\to-1)\to\infty$. We can
also build linear combinations of the functions $C_{i}(z)$, for instance

$G(z)=C_{1}(z)-2C_{3}(z)=\frac{1}{1-z}\ ,$ (B.5)

where the singularity at $z=-1$ is removed. When we now insert ${\cal
C}=C_{1}(z\to-1)=C_{3}(z\to-1)$, and treat it as a finite constant, we obtain
the finite value, which indeed matches $\zeta(0)$, as we saw in eq. (2.10),
and which Ramanujan had reported [10],

${\cal C}\stackrel{{\scriptstyle\wedge}}{{=}}-{\cal
G}\stackrel{{\scriptstyle\wedge}}{{=}}-\tfrac{1}{2}=\zeta(0)\ .$ (B.6)

It was (apparently) a step of this kind that Ramanujan performed to compute
the famous series (2.6), ${\cal R}=1+2+3+4+5\dots\ .$ Here we consider the
regularizations

$\displaystyle R_{1}(z)$ $\displaystyle=$ $\displaystyle
1-2z+3z^{2}-4z^{3}+5z^{4}\dots=C_{1}(z)^{2}=-C_{1}^{\prime}(z)=\frac{1}{(1+z)^{2}}\
,$ $\displaystyle R_{2}(z)$ $\displaystyle=$ $\displaystyle
1+2z^{2}+3z^{4}+4z^{6}\dots=\eta(z^{2})=\frac{1}{(1-z^{2})^{2}}\ .$ (B.7)

We introduced $R_{1}(z)$ before in eq. (2.7), and we used it, at the
regularized level, in identity (2.8), which we can now write in the compact
form

$\eta(z)=R_{1}(z)+4zR_{2}(z)\ .$ (B.8)

Again the singularity at $z=-1$ cancels on the right-hand side: note the an
expansion in $\varepsilon=z+1$ leads to different Laurent series for
$R_{1}(\varepsilon)=1/\varepsilon^{2}$ and
$R_{2}(\varepsilon)=(1+\varepsilon+3\varepsilon^{2}/4)/4\varepsilon^{2}+{\cal
O}(\varepsilon)$, such that the right-hand side of eq. (B.8) takes the
expected form $1/4+{\cal O}(\varepsilon)$.

If we insert ${\cal R}=R_{1}(-1)=R_{2}(-1)$, and treat ${\cal R}$ as a finite
constant, we retrieve Ramanujan’s famous result

$\frac{1}{4}\stackrel{{\scriptstyle\wedge}}{{=}}-3{\cal
R}\quad\Rightarrow\quad{\cal
R}\stackrel{{\scriptstyle\wedge}}{{=}}-\frac{1}{12}=\zeta(-1)\ .$ (B.9)

However, this procedure only works when the terms are arranged such that the
limit of interest ($z\to-1$ in our case) is regular, otherwise this step is
not controlled. Consider for instance the identity

$R_{1}(z)+C_{1}(z)=2-3z+4z^{2}-5z^{3}\dots=\frac{1}{z}[1-R_{1}(z)]\ .$ (B.10)

If we now insert ${\cal C}=C_{1}(-1)$ and ${\cal
R}=R_{1}(-1)=-\tfrac{1}{z}R_{1}(z)|_{z=-1}$, we end up with ${\cal
C}\mathop{\stackrel{{\scriptstyle?}}{{=}}}-1$, which contradicts eq. (B.6).

Another example, which refers to ${\cal R}$, is the identity

$R_{2}(z)=R_{1}(z)\eta(z)=\frac{1}{(1+z)^{2}(1-z)^{2}}\ .$ (B.11)

Carelessly inserting ${\cal R}=R_{1}(-1)=R_{2}(-1)$ purports ${\cal
R}\mathop{\stackrel{{\scriptstyle?}}{{=}}}\tfrac{1}{4}{\cal R}$, ${\cal
R}\mathop{\stackrel{{\scriptstyle?}}{{=}}}0$ or $\infty$. The reason is that
eqs. (B.10) and (B.11) are singular at $z=-1$.

Ramanujan had either the right intuition to pick an appropriate relation,
(2.8) or (B.8), where this step works, or it is based on additional
considerations on his slate, which are not documented. It seems that he hardly
knew any literature about the $\zeta$-function, but he rediscovered correct
values of its analytic continuation to $\zeta(0)$, $\zeta(-1)$ and
$\zeta(-3)$, which cannot be by accident. In particular, he must have observed
[10] the agreement of the results that he obtained in this way with the finite
term in the series that we call Euler-Maclaurin expansion, cf. Section 3.

##  Appendix C The Riemann $\zeta$-function

We have seen that a naïve ansatz for the continuation can be plagued by
subtleties when we hit a pole. An unambiguous approach to evaluate series like
${\cal C}$ and ${\cal R}$, as well as the cubic series (1.4), combines the
terms such that — in the limit of interest — the singularity is removed, as it
is done in the approaches of eqs. (2.8), (B.8), and of eqs. (2.10), (2.11), or
by subtracting the corresponding integral: the result, given in eq. (3.9),
coincides with $\zeta(-k)$, $k\in\mathbb{N}_{0}$.

The underlying concept is analytic continuation: if a complex function
$f(z)\in\mathbb{C}$, $z\in\mathbb{C}$, is holomorphic in some region, then its
analytic continuation is unique. Hence the existence of a complex derivative
is a powerful property: a plausibility argument is that such a map $f(z)$,
with $f^{\prime}(z)\neq 0$, is angle-preserving, cf. footnote 6, which
constrains the analytic continuation to a single possibility.

Being a pioneer in this field, Riemann extended the $\zeta$-function from the
region with ${\rm Re}\,z>1$ (where it is defined by the convergent series
(1.5)) to $\mathbb{C}-\\{1\\}$ by means of relations [14], which can be
condensed into the functional equation

$\zeta(z)=\frac{(2\pi)^{z}}{\pi}\sin(\frac{\pi
z}{2})\,\Gamma(1-z)\,\zeta(1-z)\ ,$ (C.1)

which is valid all over $\mathbb{C}$. We read off
$\zeta(-1)=(2\pi^{2})^{-1}(-1)\pi^{2}/6=-1/12$, where we inserted
$\Gamma(n)=(n-1)!$, $n\in\mathbb{N}_{+}$, as well as Euler’s Basel formula
$\sum_{n\geq 1}1/n^{2}=\pi^{2}/6$. Similarly we obtain
$\zeta(-3)=6\zeta(4)/(8\pi^{4})=1/120$, by employing Euler’s result
$\zeta(4)=\pi^{4}/90$. This confirms again the Ramanujan summations (1.3) and
(1.4). We further see that $\zeta(-2n)=0$, $\forall n\in\mathbb{N}$ (due to
the sin-function), in agreement with eq. (3.9).212121These are the “trivial
zeros” of $\zeta(z)$. According to the famous Riemann Conjecture, all other
(“non-trivial”) zeros have ${\rm Re}\,z=1/2$. Finally we observe a simple pole
at $z=1$, with the residue ${}^{\lim}_{z\to 1}(z-1)\zeta(z)=1$, which is
consistent with $\zeta(0)=-\tfrac{1}{2}={\cal C}$.

The validity of a series representation of $\zeta(z)$, which converges all
over $\mathbb{C}-\\{1\\}$, was demonstrated by Helmut Hasse [53]. This formula
uses eq. (1.7) and a double sum for $\eta(z)$,

$\zeta(z)=\frac{1}{1-2^{1-z}}\sum_{n\geq
0}\frac{1}{2^{n+1}}\sum_{k=0}^{n}(-1)^{k}\left(\begin{array}[]{c}n\\\
k\end{array}\right)\frac{1}{(k+1)^{z}}\ .$ (C.2)

For $z=0$ the sum over $k$ corresponds to $(1-1)^{n}=\delta_{n,0}$, and we
confirm $\zeta(0)=-1/2$. Similarly, for $z=-1$ the second sum yields
$\delta_{n,0}-\delta_{n,1}$, and we obtain once more $\zeta(-1)=-1/12$.

So could we have directly quoted these values and skipped the previous
consideration? There are two objections: we would not capture the magic of
Ramanujan’s way of thinking, and we would have missed the physical picture
which leads to the values of $\zeta(-n)$, $n\in\mathbb{N}$. This picture
justifies their application to the Casimir effect as a basic example of
renormalization.

## References

  * [1] https://mathworld.wolfram.com/
  * [2] G. S. Carr, A Synopsis of Elementary Results in Pure and Applied Mathematics: Containing Propositions, Formulae, And Methods Of Analysis, With Abridged Demonstrations, C. F. Hodgson and Son, London, 1886.
  * [3] R. Kanigel, The Man Who Knew Infinity: A Life of the Genius Ramanujan, Charles Scribner’s Sons, 1991.
K. Srinivas Rao, Srinivasa Ramanujan: A Mathematical Genius, East West Book
Pvt Ltd, 1998.

  * [4] G. S. Hardy, S. Ramanujan F.R.S., Nature 105 (1920) 494–495.
  * [5] http://ramanujan.sirinudi.org/html/published$\\_$papers.html
  * [6] G. H. Hardy and S. Ramanujan, Asymptotic Formulæ in Combinatory Analysis, Proc. London Math. Society 17 (1918) 75–115.
  * [7] S. Ramanujan, Modular equations and approximations to $\pi$, Quart. J. Math. 45 (1914) 350–372.
  * [8] A. Robinson, Film: In search of Ramanujan, Nature 531 (2016) 576–577.
  * [9] https://www.qedcat.com/misc/ramanujans$\\_$letter.jpg
  * [10] S. Ramanujan, Second Notebook (unpublished), Chapter VI.
{authentically reproduced in Ref. [11], Chapter 8}

  * [11] B. Candelpergher, Ramanujan summation of divergent series, Lectures Notes in Mathematics, 2185 [hal-01150208v2], 2017.
  * [12] https://www.youtube.com/watch?list=PLmNp3NTX4KXIqtjt2AKOr4
DUIxn64xj8G&v=jcKRGpMiVTw

  * [13] R. Ayoub, Euler and the Zeta Function, Am. Math. Mon. 81 (1974) 1067–1086.
  * [14] B. Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, in Monatsberichte der Berliner Akademie, November 1859.
A detailed discussion is included in:
H. M. Edwards, Riemann’s Zeta Function, Dover, 2001.

  * [15] J. J. Gray, Bernhard Riemann, in Encyclopedia Britannica.
https://www.britannica.com/biography/Bernhard-Riemann

  * [16] M. du Sautoy, The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, HarperCollins, 2003.
  * [17] D. J. Struik, A Concise History of Mathematics, Dover Publications, 1987 (4th edition).
  * [18] M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley Advanced Book Program, 1995.
A. Zee, Quantum Field Theory in a Nutshell, Princeton University Press, 2010.
M. Kachelriess, Quantum Fields: From the Hubble to the Planck Scale, Oxford
University Press, 2018.

  * [19] W. Bietenholz, What are Elementary Particles? From Dark Energy to Quantum Field Excitations, Rev. Cub. Fis. 37 (2020) 146–151.
  * [20] H. B. G. Casimir and D. Polder, The Influence of Retardation on the London-van der Waals Forces, Phys. Rev. 73 (1948) 360–372.
  * [21] H. B. G. Casimir, On the attraction between two perfectly conducting plates, Proc. Kon. Ned. Akad. Wetensch. B 51 (1948) 793–795.
  * [22] T. M. Apostol, An Elementary View of Euler’s Summation Formula, Am. Math. Mon. 106 (1999) 409–418.
  * [23] S. Ramanujan, Some properties of Bernoulli’s numbers, J. Indian Math. Soc. 3 (1911) 219–234.
  * [24] E. M. Lifshitz, The Theory of Molecular Attractive Forces between Solids, Soviet Phys. JETP 2 (1956) 73–83.
  * [25] C. D. Fosco, F. C. Lombardo and F. D. Mazzitelli, Vacuum fluctuations and generalized boundary conditions, Phys. Rev. D 87 (2013) 105008.
B. A. Juárez-Aubry and R. Weder, Quantum field theory with dynamical boundary
conditions and the Casimir effect, arXiv:2004.05646 [hep-th]; Quantum field
theory with dynamical boundary conditions and the Casimir effect II: Coherent
states, J. Phys. A 54 (2021) 105203.

  * [26] S. K. Lamoreaux, Demonstration of the Casimir Force in the 0.6 to 6 $\mu$m Range, Phys. Rev. Lett. 78 (1997) 5–8.
  * [27] U. Mohideen and A. Roy, Precision Measurement of the Casimir Force from 0.1 to 0.9 $\mu$m, Phys. Rev. Lett. 81 (1998) 4549–4552.
  * [28] G. Bressi, G. Carugno, R. Onofrio and G. Ruoso, Measurement of the Casimir Force between Parallel Metallic Surfaces, Phys. Rev. Lett. 88 (2002) 041804.
  * [29] S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61 (1989) 1–23.
K. A. Milton, The Casimir effect: Physical manifestatios of zero-point energy,
World Scientific, 2001\.
L. Feng, J. March-Russell, S. Sethi and F. Wilczek, Saltatory Relaxation of
the Cosmological Constant, Nucl. Phys. B 602 (2001) 307–328.
P. J. E. Peebles and B. Ratra, The Cosmological Constant and Dark Energy, Rev.
Mod. Phys. 75 (2003) 559–606.

  * [30] S. M. Carroll, The Cosmological Constant, Living Rev. Rel. 4:1 (2001) 1–56.
  * [31] V. Sahni and A. Starobinsky, The Case for a Positive Cosmological $\Lambda$-term, Int. J. Mod. Phys. D 9 (2000) 373–443.
  * [32] A. Riess et al., Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant, Astron. J. 116 (1998) 1009–1038.
S. Perlmutter et al., Measurement of $\Omega$ and $\Lambda$ from 42 High-
Redshift Supernovae, Astrophys. J. 517 (1999) 565–586.

  * [33] J. Schwinger, Casimir effect in source theory, Lett. Math. Phys. 1 (1975) 43–47. J. Schwinger, J. DeRaad and K. A. Milton, Casimir Effect in Dielectrics, Ann. Phys. (N.Y.) 115 (1978) 1–23.
  * [34] K. Scharnhorst, On propagation of light in the vacuum between plates, Phys. Lett. B 236 (1990) 354–359 [Erratum: Phys. Lett. B 787 (2018) 204].
  * [35] I. E. Dzyaloshinskii, E. M. Lifshitz and L. P. Pitaevskii, General theory of van der Waals’ forces, Sov. Phys. Uspekhi 4 (1961) 153–176.
  * [36] R. L. Jaffe, The Casimir Effect and the Quantum Vacuum, Phys. Rev. D 72 (2005) 021301.
  * [37] T. H. Boyer, Quantum Electromagnetic Zero-Point Energy of a Conducting Spherical Shell and the Casimir Model for a Charged Particle, Phys. Rev. 174 (1968) 1764–1774.
  * [38] S. G. Mamaev and N. N. Trunov, Dependence of the vacuum expectation values of the energy-momentum tensor on the geometry and topology of the manifold, Theor. Math. Phys. 38 (1979) 228–234.
J. Ambjørn and S. Wolfram, Properties of the vacuum. I. Mechanical and
thermodynamic, Annals Phys. 147 (1983) 1–32.

  * [39] J. N. Munday, F. Capasso and V. A. Parsegian, Measured long-range repulsive Casimir-Lifshitz forces, Nature 457 (2009) 170–173.
  * [40] M. P. Hertzberg, R. L. Jaffe, M. Kardar and A. Scardicchio, Attractive Casimir Forces in a Closed Geometry, Phys. Rev. Lett. 95 (2005) 250402.
  * [41] R. M. Cavalcanti, Casimir force on a piston, Phys. Rev. D 69 (2004) 065015.
  * [42] P. van Baal, The QCD vacuum, Nucl. Phys. Proc. Suppl. 63 (1998) 126–137.
  * [43] G. Bhanot, W. Fischler and S. Rudaz, A multipole expansion and the Casimir-Polder effect in quantum chromodynamics, Nucl. Phys. B 155 (1979) 208–236.
  * [44] F. Canfora and L. Rosa, Casimir energy in the Gribov-Zwanziger approach to QCD, Phys. Rev. D 88 (2013) 045025.
  * [45] O. Panella and A. Widom, Casimir effects in gravitational interactions, Phys. Rev. D 49 (1994) 917–922.
J. Q. Quach, Gravitational Casimir Effect, Phys. Rev. Lett. 114 (2015) 081104
[Erratum: Phys. Rev. Lett. 118 (2017) 139901].
J. Hua and H. Yu, Gravitational Casimir-Polder effect, Phys. Lett. B 767
(2017) 16–19.

  * [46] F. Pinto, Gravitational Casimir effect, the Lifshitz theory, and the existence of gravitons, Class. Quant. Grav. 33 (2016) 237001.
  * [47] E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko and S. Zerbini, Zeta Regularization Techniques with Applications, World Scientific, 1994.
  * [48] G. H. Hardy and J. E. Littlewood, Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes, Acta Mathematica 41 (1916) 119–196.
  * [49] S. W. Hawking, Zeta function regularization of path integrals in curved spacetime, Commun. Math. Phys. 55 (1977) 133–148.
  * [50] J. Polchinski, String Theory: Volume 1, An Introduction to the Bosonic String, Cambridge Monographs on Mathematical Physics, 1998.
F. Toppan, Lecture Notes: String Theory and Zeta-function, CBPF-MO-002/00,
http://www.cbpf.br/$\sim$toppan/mo00200.pdf

  * [51] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, Section 15.3.5.
  * [52] B. Zwiebach, A First Course in String Theory, Cambridge University Press, 2004, Section 12.4.
  * [53] H. Hasse, Ein Summierungsverfahren für die Riemannsche Zeta-Reihe, Math. Z. 32 (1930) 458–464.