# Accurate first-principles bandgap predictions
in strain-engineered ternary III-V semiconductors

Badal Mondal  Wilhelm-Ostwald-Institut für Physikalische und Theoretische
Chemie, Universität Leipzig, 04103 Leipzig, Germany  Fachbereich Physik,
Philipps-Universität Marburg, 35032 Marburg, Germany  Marcel Kröner  Material
Science Center and Department of Physics, Philipps-Universität Marburg,
D-35043 Marburg, Germany  Thilo Hepp  Material Science Center and Department
of Physics, Philipps-Universität Marburg, D-35043 Marburg, Germany  Kerstin
Volz  Material Science Center and Department of Physics, Philipps-Universität
Marburg, D-35043 Marburg, Germany  Ralf Tonner-Zech  ralf.tonner@uni-
leipzig.de Wilhelm-Ostwald-Institut für Physikalische und Theoretische Chemie,
Universität Leipzig, 04103 Leipzig, Germany

###### Abstract

Tuning the bandgap in ternary III-V semiconductors via modification of the
composition or the strain in the material is a major approach for the design
of optoelectronic materials. Experimental approaches screening a large range
of possible target structures are hampered by the tremendous effort to
optimize the material synthesis for every target structure. We present an
approach based on density functional theory efficiently capable of providing
the bandgap as a function of composition and strain. Using a specific density
functional designed for accurate bandgap computation (TB09) together with a
band unfolding procedure and special quasirandom structures, we develop a
computational protocol to predict bandgaps. The approach’s accuracy is
validated by comparison to selected experimental data. We thus map the bandgap
over the phase space of composition and strain (we call this the “bandgap
phase diagram”) for several important III-V compound semiconductors: GaAsP,
GaAsN, GaPSb, GaAsSb, GaPBi, and GaAsBi. We show the application of these
diagrams for identifying the most promising materials for device design.
Furthermore, our computational protocol can easily be generalized to explore
the vast chemical space of III-V materials with all other possible
combinations of III- and V-elements.

III-V semiconductors, ternary, strain, bandgap, direct-indirect transition,
bandgap phase diagram

††preprint: APS/123-QED

##  I Introduction

Materials based on III-V semiconductor compounds are attracting much attention
in science and engineering due to their diverse applications in fields such as
optoelectronics [1, 2]. One of the main goals of basic and applied research is
to tailor materials’ optical properties to a specific application [3, 4, 5, 6,
7, 8]. One of the most critical fundamental properties in this respect is the
bandgap, both in terms of size and type (direct or indirect). For example,
optical telecommunication applications require materials with direct bandgaps
in the range of 0.80–0.95 eV [3, 4, 5], while solar cell applications require
a range of 0.5–2.0 eV [6, 7, 8]. Composition engineering, i.e., changing the
relative composition of group 13 and 15 elements in ternary III-V compounds,
is one of the most important approaches to adjusting the bandgap [9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Systematic application of strain
such as mechanical strain (e.g., external pressure [23, 24, 25, 26],
mechanical bending of nanowires [27, 28, 29]) or strain due to lattice
mismatch (e.g., core-shell mismatch in nanowires [30, 31, 32, 33, 34]) on a
system are alternative strategies to tailor the bandgap. Combining composition
and strain engineering, the bandgap can be tuned over a wide range of values,
and direct or indirect semiconductors can be designed. In thin-layer
heteroepitaxy, choosing the substrate-layer combination with minimum lattice
mismatch is often desirable to minimize the strain effect from the substrate.
However, in practice, perfect lattice matching is rarely possible. In such
cases, not only the composition but the effect of inherent strain from the
substrate also substantially affects the active layer’s bandgap [9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 35, 36, 37, 38, 39]. Therefore,
one requires a complete knowledge of the material-specific dependence of the
bandgap on composition and strain to guide the optimal choice of materials.
However, exploring the vast chemical space of all possible combinations of
III- and V-elements with variation in composition and strain is experimentally
not feasible. Additionally, growing a new material is often challenging
because of thermodynamic or kinetic limitations, such as phase separation or
surface roughening, in addition to the demanding task of optimizing the growth
conditions [9, 13, 16, 19, 21, 22]. This makes an experimental screening
approach of vast compound and strain spaces unrealistic. We thus aim in this
study to develop a reliable and predictive theoretical approach.

Two major theoretical approaches that have been used to analyze strain effects
on the bandgap of III-V materials are (semi-)empirical methods and ab initio
approaches. Although (semi-)empirical methods such as k$\cdot$p theory [36,
40] and tight-binding methods [36, 41, 42, 43] are computationally efficient,
they rely on empirical parameters which require system-specific experimental
input data. This strongly limits the predictive ability of these methods for
new or yet unknown materials. Additionally, in case of a large mismatch in
atomic sizes of the constituting elements, the ternary material shows local
strain effects, severely affecting the bandgap [44]. These local strain
effects, however, can not be included in empirical approaches and, hence, are
neglected. Then again, ab initio approaches such as density functional theory
(DFT) [40, 45, 46, 47, 48, 49, 50] allow for the calculation of electronic
properties from first-principles and are thus predictive if accurate density
functional are used. The relaxation of the atomic positions also allows to
properly include and investigate the effect of the local strain on the
electronic properties in these approaches. Additionally, recent advancements
in the modeling strategies of alloy systems using quasirandom supercells [51,
52, 53, 54, 55] allow for electronic properties calculations in the ab initio
approaches, even for diluted and disordered materials. An accurate alternative
to DFT approaches is the use of GW-based methods, which are nevertheless too
computationally demanding for screening approaches as intended here [44, 45,
46, 56].

In a previous study, we established a computational protocol for predictive
modeling based on DFT for binary III-V compounds over a wide range of strain
values [45]. In this study, we are now extending this approach to ternary
III-V compounds, which then allow the combination of strain and composition to
fully explore a bandgap design approach. For ternary systems, only the effects
of composition variations on the bandgap in unstrained materials have been
studied [40, 41, 44, 51, 52, 53, 54, 55]. For strained materials, a suitable
theoretical framework is still lacking. We present here a predictive first-
principles protocol for a complete mapping of the mutual correlation of
composition, strain, and bandgap in ternary III-V semiconductor systems. The
goal is to provide guidelines for assessing and identifying the most promising
target materials for experimental investigations in the future.

We start by describing the computational methods in Sec. II. Next, we describe
the protocol for determining the nature of the bandgap from supercell
calculations using GaAsP as an example in Sec. III. We further present the
composition-strain-bandgap correlation results for different ternary III-V
semiconductors in Sec. IV. We start with GaAsP, an experimentally well-studied
and promising candidate for LEDs, detectors, and Si-based multijunction solar
cells [57, 58, 59, 60, 61, 62, 63, 64]. The results for the GaAsN compound, a
promising laser-active material [44, 65, 66, 67], are presented next. To show
the general applicability of our approach, we then show selected results for
(i) GaPSb, a candidate for vertical cavity emitting surface laser [68, 69, 70,
71, 72]; (ii) GaAsSb, a material for tandem solar cell application [73, 74];
(iii) GaPBi, a promising material for near-infrared photonic device
application on Si [75, 76]; and (iv) GaAsBi, another material discussed for
near- and mid-infrared photonic device application [77, 78, 79]. We then
discuss the comparison of our computations with experimental data in Sec. V,
underlining the accuracy and predictive capability of our computational
approach.

##  II Computational details

The calculations were performed with DFT-based approaches as implemented in
the Vienna ab initio simulation package (VASP 5.4.4) [80, 81, 82, 83], using
plane wave basis sets in conjunction with the projector-augmented wave (PAW)
approach [84, 85]. The ternary materials were generated using the special
quasirandom structures (SQS) approach [86] with a supercell of size $6\times
6\times 6$. The SQS cells were generated using the alloy theoretic automated
toolkit (ATAT) [87, 88, 89]. For all the materials except GaAsN, one SQS cell
was used per composition. In GaAsN, in agreement with the previous observation
[44], we found that the size of the bandgap strongly depends on the
distribution of N atoms in the supercell, even in the SQS approach. We thus
used 10 SQS cells for each composition in this case.

Geometry optimization of the supercells was performed using the PBE functional
[90], including the dispersion-correction method DFT-D3 with an improved
damping function [91, 92]. The basis set energy cut-off was set to 450 eV. The
electronic energy convergence criteria of $10^{-6}$ eV and the force
convergence of $10^{-2}$ eVÅ-1 were used. The reciprocal space was sampled at
the $\Gamma$-point only, given the large supercells used [93]. The meta-GGA
functional TB09 [47] was used to calculate the electronic properties (bandgaps
and band structures). The effects of spin-orbit coupling were considered in
the TB09 calculations. For the meta-GGA calculations, the energy cut-off of
the basis set and the convergence criterion for the electronic energy were
lowered to 350 eV and $10^{-4}$ eV, respectively, to reduce the computational
costs. Structure optimizations were carried out by consecutive volume and
position optimization until convergence was reached. This setup was previously
used to generate bandgaps in excellent agreement with experimental data [44].

All the materials within the composition range investigated here feature the
zincblende-type structure only. Moreover, [100] crystal direction is the most
common choice of substrate orientation and growth direction in epitaxy.
Therefore, we modeled the strain application along [100] directions only. The
isotropic strain was modeled by increasing (decreasing) all the lattice
parameters of the unstrained structure by the same amount. In this case, only
the atomic positions of the strained structure were optimized, keeping the
volume fixed. For biaxial strain, the in-plane lattice parameters were kept
fixed, and the lattice parameter in the out-of-plane direction was optimized.
No structural phase transition is assumed under strain application. More
details on the strain modeling can be found in Ref. [45]. In the following, we
indicate tensile strain with a positive sign and compressive strain with a
negative sign.

DFT calculations were performed at discrete points in composition-strain space
(Fig. S6 [94]). The calculated bandgap values were then interpolated to create
the final images in Figs. 2, 3, 4, 5 and 6. Noticeably, for the systems we
addressed in this article, the variations of bandgap values with concentration
and strain are mostly non-monotonic (Fig. S6 [94]). This resulted in non-
smooth interpolation in Figs. 2, 3, 4, 5 and 6. It is to be stressed that the
origin of the non-smooth patterns is neither an interpolation artifact nor a
deficiency of our DFT protocol. This solely originated because of the non-
monotonic variation of the bandgap values (in the composition-strain space) of
the SQS cells that we used to calculate bandgaps. A choice of positive
smoothening during interpolation (e.g., bivariate B-spline, gaussian
filtering) could mitigate the problem but significantly increased the
deviation of the interpolated bandgap values from the calculated DFT values
and was thus not chosen. Further detail of the interpolation procedures can be
found in Sec. S VI [94]. Moreover, the nature of bandgaps can solely be
deduced from the direct-indirect transition lines and thus requires no
interpolation.

##  III Protocol for determining bandgap nature

(a)

(b)

Figure 1: Variation of the bandgap under isotropic compressive strain for
GaAs0.963P0.037. The $\Gamma$-, L-, and X-BSW of the folded supercell
conduction band are given in parentheses in the format ($\Gamma$:L:X). The
vertical lines in (a) separate regions where the CBM changes character. In
(b), the strain resolution is increased to determine the point of direct-
indirect transition more accurately, indicated by the red circle (where the
highest BSW changes from $\Gamma$ to L).

Supercell calculations, as required for modeling ternary semiconductors, lead
to the folding of band structures [104, 105]. The size of the bandgap can be
well extracted from the folded band structure, which represents the energy
difference between the highest occupied VB and the lowest unoccupied CB
obtained from supercell calculations (folded bands). However, determining the
bandgap’s nature requires the primitive Bloch character of the bands to be
known, which gets mixed up in the supercell eigenstates. With the band
unfolding method, one projects these supercell eigenstates on the eigenstates
of a suitable reference primitive cell. This requires the calculation of Bloch
spectral weights (BSW), which measure the fraction of the primitive Bloch
character in a supercell eigenstate. The result is an effective band structure
(EBS) [51, 52, 53, 54, 55]. The spectral weights, $w_{n,{\bf K}}\left({\bf
k}\right)$, can be calculated from the plane wave coefficients as described in
Ref. [55]:

$w_{n,{\bf K}}\left({\bf k}_{j}\right)=\sum_{\bf g}|C_{n,{\bf K}}\left({\bf
g}+{\bf G}_{j}\right)|^{2}$ (1)

where $n$ represents the band index, and the reciprocal lattice vectors of the
primitive and supercell are denoted by ${\bf g}$ and ${\bf G}_{j}$,
respectively. The index $j$ accounts for the series of primitive vectors,
${\bf k}_{j}={\bf K}+{\bf G}_{j}$. The code “fold2Bloch” from Ref. [54] was
used to calculate the BSW values.

In our previous study on binary III-V systems [45], we have shown that the
valence band maxima (VBM) always remain at the $\Gamma$-point, and only the
conduction band minima (CBM) change their position in reciprocal space under
strain. We have also shown that the CBM occurs only at the $\Gamma$-, L-, and
(near) X-point in the band structure under strain. Therefore, it is sufficient
to trace the conduction band (CB) at these points to determine the nature of
the bandgap. As in our previous study on binary systems, we focus here on
analyzing ternary III-V compounds with zincblende structures. For these
structures in the $6\times 6\times 6$ supercell dimensions chosen here, the
$\Gamma$-, L- and X-point of the primitive band structure fold to the
$\Gamma$-point in the supercell [53, 54, 55]. Therefore, it is sufficient to
calculate the BSWs of solely the CB at the $\Gamma$-point in the supercell
calculation to determine the nature of the bandgap. Consequently, we performed
the supercell calculations by sampling the reciprocal space only at the
$\Gamma$-point and unfolded the CB.

Figure 1 shows the steps for determining the bandgap nature from supercell
calculations more clearly. Figure 1a shows the bandgap variation for GaAsP
with 3.7% P concentration under isotropic compressive strain. The $\Gamma$-,
L-, and X-BSWs of the folded supercell CB are given in parentheses. This shows
100% $\Gamma$-BSW for the unstrained structure in line with the direct
bandgap. With increasing strain, the $\Gamma$-BSW decreases (first number in
brackets), and the L-BSW increases (second number in brackets). After a
certain amount of strain, the L-character of the CB dominates. The bandgap
becomes indirect in nature. Notably, once the strain values reach the point of
direct-indirect transition (DIT) in the bandgap nature (around –1.5% strain),
the bandgap values begin to decrease further with additional strain. This
trend is like what we previously observed in binary III-V semiconductor
systems, where a strong dependence of the band energies ($E$) on the
wavevectors (k) under strain was found, leading to a non-monotonic variation
of bandgap values with strain [45]. Moreover, we found that such non-monotonic
behaviour in bandgap values under strain points to a DIT [45]. In ternary
III-V semiconductor systems, we have now found that similar non-monotonic
behaviour in bandgap values under strain also indicates a DIT. Further
compressing the system then leads to a transition of the CB character from L
to X.

In Fig. 1b, we show calculations with increased resolution in strain to
accurately determine the transition to L corresponding to the sought point of
DIT at $-1.4$% strain. We define the last strained structure with bandgap of
direct nature before the transition to the indirect bandgap as the transition
point (the red circle in Fig. 1b). In the Supplemental Material (Fig. S1
[94]), we have given the EBSs of GaAs0.963P0.037 for different strain values.
These confirm our analyses.

If the difference in BSW between different points in k-space is large, the
nature of the bandgap can be unanimously determined. However, close to the
transition points, in some cases, the differences are more subtle (Fig. S2
[94]). We, therefore, set a cut-off criterion of 20% BSW. If the $\Gamma$-BSW
is larger than the cut-off criterion, then the direct transition has a finite
probability even if the L- or X-BSW is larger than the $\Gamma$-BSW. In such
cases, the bandgap is called “partially direct”. This defines a “region of
uncertainty” in the bandgap nature. We chose the 20% cut-off criterion because
this produces results that agree best when compared to the experiments for
several systems. For the GaPBi system, however, the 10% BSW cut-off criterion
produces the best agreement.

In some systems such as GaAsN, the band originating from the added nitrogen
atoms, the so-called “defect N state” [54, 106, 107, 108, 109], is strongly
dispersed under strain (Figs. S3b and S4b [94]). Therefore, we set another
cut-off criterion of 20% BSW as a minimum limit for a (defect) eigenstate to
be considered an eigenstate (Fig. S5 [94]). Starting from the lowest
unoccupied CB, we search for eigenstates until the cut-off BSW criterion is
met, at which point we consider it to be the redefined CB. If none of the CBs
satisfy the cut-off criterion, we use the lowest CB for determining the
bandgap nature. Accordingly, in these cases, we calculate the bandgap values
as the energy difference between the highest VB and the redefined CB. When
redefining, unoccupied CB states that do not satisfy the cut-off criteria are
disregarded. This led to an increase in the bandgap values, as is observed in
Fig. S5 [94].

##  IV Results

In this section, we present the bandgaps calculated for different materials
and determine their nature according to the above protocol. We mapped the
bandgaps in terms of their size and nature for various strained ternary III-V
compounds. We start with two important ternary III-V semiconductor materials,
GaAsP and GaAsN. Then we show selected data for the material systems GaPSb,
GaAsSb, GaPBi, and GaAsBi.

###  IV.1 GaAsP

Figure 2: Isotropic strain for GaAsP. The variation of bandgap magnitudes
(E${}_{\text{g}}$) and type as a function of composition and strain. The
dashed black horizontal line indicates unstrained GaAsP. The black circles are
the calculated DIT points. The direct and indirect enclosed regions describe
the nature of bandgap being direct and indirect, respectively. The hatched
pattern region is the “uncertainty region” (see Sec. III).

For the case of isotropic strain, Fig. 2 shows the bandgap as a function of
composition ($x=$ 0–100% in GaAs1-xPx) from 5% tensile to 5% compressive
strain. The bandgap value varies between 0.32–2.42 eV in the strain regime
investigated. For the same amount of P concentration, the bandgap primarily
increases in moving from tensile to compressive. Furthermore, the figure shows
that in going from compressive to tensile strain, the DIT occurs at a higher
concentration of P atoms. The dashed horizontal line marks the data
corresponding to the unstrained structures for different fractions of P. The
intersection of this line with the DIT line shows at which percentage of
phosphorous contribution the unstrained structure shows a DIT. This transition
occurs at $x=37$%. Here, the bandgap shows a value of 1.96 eV. The terms
direct and indirect in the figure correspond to the area where the bandgap is
direct and indirect, respectively. Due to the similarity with commonly used
phase diagrams, we call this representation a “bandgap phase diagram”. This
and the following figures thus provide a 2D representation of the bandgap
phase diagram for the ternary materials.

For the biaxial strain regime, Fig. 3 shows the bandgap phase diagram for
GaAsP as a function of composition from 5% tensile to 5% compressive strain.
The value of the bandgap varies in a range of 0.82–2.42 eV. For the same
amount of P atoms, the bandgap reaches a maximum around the unstrained
structure and gets smaller for tensile as well as compressive strain. This is
different from the isotropic strain case. For unstrained GaP, the bandgap
value is 2.36 eV. The nature of the bandgap also shows a different trend
compared to Fig. 2. The range of strain around the unstrained structure where
a direct bandgap is found gets smaller for higher amounts of P. This is in
line with GaAs (0% P) being a direct and GaP (100% P) being an indirect
semiconductor. The largest amount of P concentration where a direct
semiconductor is found is 39–40% P in the unstrained structure. This is
similar to the previous experimental result (45% P) [10, 110].

Figure 3: Biaxial strain for GaAsP. The variation of bandgap magnitudes
(E${}_{\text{g}}$) and type as a function of composition and strain. The
dashed black horizontal line indicates unstrained GaAsP. The black circles are
the calculated DIT points. The DIT points are fitted with a 5th-order
polynomial. The direct and indirect enclosed regions describe the nature of
bandgap being direct and indirect, respectively. The hatched pattern region is
the “uncertainty region” (see Sec. III). Solid black lines indicate the
substrate lines under “epitaxial growth” model.

One of the most common approaches to experimentally realize biaxial strain in
III-V semiconductors is epitaxial growth. As pointed out in Ref. [45], biaxial
strain can be used to model epitaxial growth. We thus investigate the effect
of different substrates in our bandgap phase diagram (Fig. 3), where each
solid line corresponds to one substrate: GaAs, GaP, InP, or Si. These solid
lines indicate how much biaxial strain would develop in the GaAsP system as
the respective value of % P when grown on the respective substrates under
idealized conditions. The (substrate) strains are calculated according to Eq.
2:

$\text{Substrate strain}(\%)=\frac{a_{sub}-a}{a}\times 100$ (2)

where $a_{sub}$ is the equilibrium lattice parameters of the substrates, and
$a$ is the lattice parameters of unstrained GaAsP systems at their respective
P concentrations. E.g., for 100% P, the strain on the GaP substrate is zero,
while growing GaAs (0% P) on GaP would result in 3.8% in-plane compressive
strain. This, of course, neglects defect formation and strain relaxations and
assumes perfect epitaxial growth. Clearly, by choosing different substrates,
the nature can be changed, and the size of the bandgap can be tuned over a
wide range. We refer to the next section for a comparison of our calculations
to experimental data.

###  IV.2 GaAsN

Figure 4: Isotropic strain for GaAsN (up to 12% N). The variation of bandgap
magnitudes (E${}_{\text{g}}$) and type as a function of composition and
strain. The dashed black horizontal line indicates unstrained GaAsN. The black
circles are the calculated DIT points. Beyond 7% N, the DIT is outside the
investigated strain regime. 10 SQS cells are used for each configuration and
strain point. The bandgaps plotted are the average bandgaps. The error bars
indicate the standard deviation in DIT points estimation. The direct and
indirect enclosed regions describe the nature of bandgap being direct and
indirect, respectively. Figure 5: Biaxial strain for GaAsN (up to 12% N). The
variation of bandgap magnitudes (E${}_{\text{g}}$) and type as a function of
composition and strain. The dashed black horizontal line indicates unstrained
GaAsN. 10 SQS cells are used for each configuration and strain point. The
bandgaps plotted are the average bandgaps. All the bandgaps are direct in
nature. Solid black lines indicate the substrate lines under the “epitaxial
growth” model.

As the next material, we investigate GaAsN. First, we show results for
isotropic strain, which results in the bandgap phase diagram shown in Fig. 4.
The results are markedly different from GaAsP, and the data set is much more
limited. In this case, we found a strong dependency of the bandgap on the N
atoms distribution in the supercell [44]. We thus used 10 SQS cells for each
data point in the figure and averaged the resulting bandgaps. This results in
an error bar for the DIT points, which is rather large for medium amounts of
nitrogen atoms due to the formation of small clusters and chains. Calculations
were only possible for up to 12% N. For higher concentration and/or high
compressive strain, our chosen supercell is not large enough to avoid the
unphysical electronic interaction of N atoms with their images in the periodic
boundary condition approach. This effect has already been discussed in Ref.
[44]. For the strain and composition regions where computation was possible,
an indirect gap is only found for low values of % N and rather large
compressive strain values. The EBSs for selected % N and strain values are
shown in Figs. S3 and S4 [94].

For biaxial strain in GaAsN, the data are shown in Fig. 5. In contrast to
GaAsP, the bandgap gets smaller with the increasing amount of nitrogen in the
system, from 1.47 eV for the unstrained case of GaAs to 0.10 eV for the highly
strained systems with a large number of N atoms. All bandgaps computed are
direct. Epitaxial growth on GaAs is reasonably possible for moderate strain
values and results in a variation of bandgap from 1.47 eV to 0.45 eV. For GaP
and Si substrates, a large strain would be exerted on the system, and mostly
lower bandgap values are found.

###  IV.3 GaPSb, GaAsSb, GaPBi, GaAsBi

(a) GaPSb

(b) GaAsSb

(c) GaPBi

(d) GaAsBi

Figure 6: Bandgap phase diagram for ternary III-V semiconductors GaEY (E= P,
As; Y = Sb, Bi) under biaxial strain. The bandgap magnitudes
(E${}_{\text{g}}$) are shown in the color bar. The dashed black horizontal
line indicates unstrained structures. The black circles are the calculated DIT
points. The direct and indirect enclosed regions describe the nature of
bandgap being direct and indirect, respectively. The hatched pattern region is
the “uncertainty region” (see Sec. III). Solid black lines indicate the
substrate lines under the “epitaxial growth” model.

The approach outlined here can be extended to other combinations of elements
in III-V semiconductor materials. Exemplarily, we present the bandgap phase
diagrams for four other important ternary compounds in Fig. 6. Since epitaxial
growth is the most interesting experimental realization method for these
compounds, we only present the data for biaxial strain.

For all compounds investigated, we find a bandgap range of 0.00–2.36 eV, with
the largest values found for the host materials GaP and GaAs in the unstrained
case. The alloys with Sb could be investigated over the full range of 0–100%
Sb in GaPSb and GaAsSb (Figs. 6a and 6b). We find a DIT for unstrained GaPSb
at 30% Sb concentration (Fig. 6a). For this compound, the DIT is shown as a
region, including the uncertainty in determining the nature of the bandgap, as
outlined in Sec. III. With the increase in the Sb fraction, the strain at
which the DITs take place increases. For GaAsSb and GaAsBi, the bandgap is
direct throughout the range investigated (Figs. 6b and 6d), while it is
indirect for GaPBi (Fig. 6c). Notably, we only investigate the bismides up to
a fraction of 15% Bi. The reason is that, similar to GaAsN (Fig. S4 [94]), for
structures with large Bi content, the strongly dispersed bands decrease the
reliability in the determination of bandgap nature. Additionally, GaPBi and
GaAsBi become metallic for higher Bi fractions. Although we find no transition
within 15% Bi, it can not be excluded that the DIT appears at higher
percentages of bismuth.

Again, we indicate the strain values associated with different typical
substrates for epitaxial growth by solid black lines in the figures. The data
show that deviating from the substrate-layer lattice-matching condition
quickly leads to high strain, and defects are highly likely to occur during
growth. Also, Si can be used as a substrate for GaPSb and GaPBi epitaxial
growth if the Sb or Bi content is not too large. The epitaxial growth of the
respective GaAs-based materials (GaAsP, GaAsN, GaAsSb, GaAsBi) will give rise
to high strain on Si-substrate throughout the whole composition region. A
noticeable change in the slope in substrate lines is found close to 10% Bi and
40% Sb concentration in GaAsBi and GaAsSb, respectively. Although we did not
find any structural phase transition in those regions, the origin of the
change in the slope is not clear to us yet.

From the above discussions, it becomes clear that bandgap phase diagrams can
be a valuable aid in deciding which substrates are good choices for targeting
a specific bandgap size and nature for a given ternary material. And vice
versa, which material to grow for a specific application and a given
substrate? We will discuss this further in the next section.

##  V Discussion

All data were derived from DFT computations to this point. In Table 1, we now
compare our calculated bandgaps with experimental data from measurements on
heteroepitaxial layer structures. The GaAsP/GaAs samples were grown by low-
pressure hydride vapor phase epitaxy (LP-HVPE). Further details can be found
in Ref. [111]. The remaining samples were grown by metalorganic vapor phase
epitaxy (MOVPE). The details of the growth characteristics of the MOVPE
samples can be found in Refs. [22, 72, 112, 113, 114, 115, 116, 117].
Experimentally, the layer thickness and bandgaps of the MOVPE samples were
determined using X-ray diffraction and room-temperature photoluminescence (RT-
PL), respectively. Except for GaPSb samples from Ref. [72], in which cases,
the PL were measured at 10 K.

Table 1: Comparison of the calculated bandgaps for investigated ternary III-V semiconductors under biaxial strain with experiments. The experimental data are for the heteroepitaxial layer structures, and the bandgaps are determined from photoluminescence (PL) measurements. The “I” in the brackets indicate the indirect nature of the bandgap. The remaining bandgaps are direct. (RT-PL: room temperature PL) |  |  | Layer | Bandgap (eV) | Deviation | Percentage | RMSD
---|---|---|---|---|---|---|---
System | Substrate | $x(\%)$ | thickness (nm) | Calculated | Experiment | (eV) | deviation (%) | (eV)
GaAs1-xPx | GaAs | 18.0 | 9000 | 1.65 | 1.66 | 0.01 | 0.6 | 0.01
25.0 | 12250 | 1.72 | 1.72 | 0.00 | 0.0
[111] | 28.0 | 13000 | 1.75 | 1.76 | 0.01 | 0.6
GaP1-xSbx111For GaPSb samples, no specific thicknesses were reported in the reference. | GaP | 14.0 | $-$ | 1.66(I) | 1.61(I) | $-0.05$ | $-3.1$ | 0.10
| GaAs | 29.0 | $-$ | 1.33(I) | 1.39(I) | 0.06 | 4.3
| 32.0 | $-$ | 1.30 | 1.31 | 0.01 | 0.8
[72] | 37.0 | $-$ | 1.24 | 1.33 | 0.09 | 6.8
| GaSb | 93.0 | $-$ | 0.56 | 0.74 | 0.18 | 24.3
GaAs1-xSbx | GaAs | 5.5 | 46.3 | 1.22 | 1.34 | 0.12 | 9.0 | 0.13
7.0 | 51.2 | 1.17 | 1.31 | 0.14 | 10.7
GaAs1-xBix | GaAs | 0.9 | 75.0 | 1.28 | 1.33 | 0.05 | 3.8 | 0.10
1.9 | 67.0 | 1.19 | 1.26 | 0.07 | 5.6
2.9 | 60.0 | 1.10 | 1.20 | 0.10 | 8.3
3.2 | 59.0 | 1.08 | 1.18 | 0.10 | 8.5
3.8 | 54.0 | 1.04 | 1.14 | 0.10 | 8.8
[112, 113, 115] | 4.8 | 25.0 | 0.98 | 1.11 | 0.13 | 11.7
[116, 117, 118] | 5.3 | 50.0 | 0.95 | 1.07 | 0.12 | 11.2
| 6.0 | 25.0 | 0.91 | 1.04 | 0.13 | 12.5
GaAs1-xNx | GaAs | 1.2 | 6.3 | 1.20 | 1.25 | 0.05 | 4.0 | 0.12
2.0 | 17.0 | 1.10 | 1.16 | 0.06 | 5.2
2.3 | 7.0 | 1.06 | 1.17 | 0.11 | 9.4
[10, 44, 114] | 2.9 | 7.0 | 1.00 | 1.11 | 0.11 | 9.9
| 5.0 | 4.0 | 0.82 | 1.01 | 0.19 | 18.8
GaAs1-xSbx | GaAs | 27.8 | 3.7 | 0.67 | 1.10 | 0.43 | 39.1 | 0.42
28.0 | 4.1 | 0.66 | 1.07 | 0.41 | 38.3
GaAs1-xNx | GaP | 4.9 | 6.0 | 0.66 | 1.18 | 0.52 | 44.1 | 0.66
| Si | 6.9 | 5.5 | 0.47 | 1.21 | 0.74 | 61.2
| 8.9 | 5.5 | 0.46 | 1.17 | 0.71 | 60.7
[10, 44, 114] | 9.5 | 6.0 | 0.45 | 1.11 | 0.66 | 59.5
| 10.9 | 5.4 | 0.45 | 1.11 | 0.66 | 59.5
GaP1-xSbx | Si | $\leq 17.5$ | 7–9 | 1.91–1.36(I) | No RT-PL observed |
GaP1-xBix | GaP | $\leq 12.0$ | 17–73 | 2.03–1.18(I) | No RT-PL observed |
[112] |

The comparison of the experimental bandgaps with our computed results shows
good agreement. The deviation is determined with respect to the root-mean-
square deviation (RMSD) from all available experimental samples. For most
structures, the RMSD is around 0.1 eV. Most computed values deviate by less
than 10% from the experimental values (exceptions are discussed separately);
in the case of GaAsP, the deviation is even more accurate ($<$1%). This
confirms our previous findings on unstrained structures that the DFT protocol
we developed gives excellent agreement to experimental bandgaps [11, 40, 44,
45]. In this study, we show that it is also applicable to compound
semiconductors under strain. For samples with very small layer thickness, the
matching of experiment and computation is less good. This can be observed for
GaAsSb/GaAs thin samples with RMSD of ca. 0.4 eV. We attribute this to the 2D
quantum confinement effect, which is found for thin samples. This confinement
effect leads to an increase in the bandgap with respect to thicker samples
[119]. This effect is not captured in our computational model as the
calculations were performed for 3D periodic strained structures. The large
deviation observed for GaAsN/GaP and GaAsN/Si samples can not be explained by
this effect alone, though. An additional effect here is the strong dependency
of the bandgap on the distribution of N atoms which has been found for
unstrained GaAsN before [44]. The dependency is further amplified under large
strain (around 3%, see Fig. 5) in those samples. In the case of GaAsN/GaAs
samples, where the N concentration investigated was around 1–5%, the strain is
relatively small ($<$1%), resulting in better agreement with the experiment as
compared to the GaAsN/GaP and GaAsN/Si samples. No RT-PL was observed for
GaPSb/Si and GaPBi/GaP samples.

(a)

(b)

(c)

(d)

Figure 7: Proposals on how the bandgap phase diagram of biaxially strained
GaAsP can be used in designing optoelectronic devices. (a) Defines the bound
of composition region for creating a QWH with direct bandgap GaAsP on GaAs
substrate. (b) Choosing the different composition regions appropriately to
make a multijunction photovoltaic with successive direct and indirect cells on
the GaAs substrate. (c) In the vicinity of the transition point, the bandgap
properties of the GaAsP epilayer on the GaP substrate can be changed by
appropriately varying the composition. (d) Depending on the choice of
substrate, GaAs or Si, the particular composition indicated by the vertical
line can be made direct or indirect bandgap, respectively.

This is consistent with our findings that those materials show indirect
bandgaps (Figs. 6a and 6c). Experimental measurements of the magnitude of the
indirect bandgaps are not available yet.

The consistent agreement between the experiment and calculated bandgaps (both
in magnitude and in nature) suggests that we are able to quantitatively
predict the bandgap over a wide range of compounds, compositions, and strain
regions. However, as discussed above, the effect of 2D confinement is also
crucial for relatively thin quantum well heterostructures and, hence, needs
further investigation.

Finally, based on the bandgap phase diagram, we propose several design
strategies to optimize the selection of material combinations for achieving
specific optical applications and new design principles for devices (Fig. 7).

In Fig. 7a, we propose a quantum-well heterostructure (QWH) composed of
biaxially strained GaAsP on GaAs substrate. As the QW layers are made out of a
single material with varied composition only, the epitaxial growth could be
performed efficiently. The bandgap phase diagram shows the areas in
compositional phase space where a direct bandgap in GaAs1-xPx can be achieved
($x<$ 34%). For $x>$ 35%, the bandgaps are indirect and hence, are
inappropriate for the heterostructure.

Figure 7b shows an efficient approach for the monolithic integration of
multiple QWH to construct multijunction photovoltaics. In this case, the QWHs
are separated by thin indirect bandgap layers of the same material as QWH but
only with a different composition. This would make the integration approach
efficient, as no sample transfer is required during growth.

In Fig. 7c, we propose a device with a gradual change in the bandgap
properties. The concept utilizes the continuous transition in the nature of
bandgap with alloy concentration in the vicinity of the DIT region. At the
amount of P chosen here ($x=$ 15–35%), we propose to grow the GaAsP epitaxial
layer on GaP with P concentration continuously changing from the direct to
indirect bandgap region or vice versa. This way, changes in the bandgap
magnitude, as well as the nature of the bandgap, are possible. Note that the
concentration gradient can be implemented both in the horizontal and vertical
directions.

Figure 7d shows another application of this concept. By appropriately choosing
the substrate, we can tune the epitaxial layer (here: GaAsP) to show either a
direct or indirect bandgap. Depending on the substrate, GaAs, or Si, the
particular composition indicated by the vertical line will show direct or
indirect bandgap, respectively.

##  VI Summary

Using density functional theory and the concept of band unfolding, we
developed a first-principles computational protocol for the comprehensive
mapping of the bandgap magnitude and type over a wide range of composition and
strain values for several ternary III-V semiconductors. We constructed the
composition-strain-bandgap relationship, the bandgap phase diagram, for
several ternary III-V semiconductors: GaAsP, GaAsN, GaPSb, GaAsSb, GaPBi, and
GaAsBi. We showed that this way of mapping the effect of strain could be used
to choose application-specific best-suited material systems and hence, is
highly beneficial to device design. In addition, we developed an efficient
approach based on Bloch spectral density for determining the nature of bandgap
from supercell calculation. Notably, our computational protocol can be
generalized to explore the vast chemical space of III-V materials with all
other possible combinations of III- and V-elements. The comparison to
experimental bandgap data underlines the accuracy of the computational
approach chosen. This approach will be extended to more complex materials in
the future.

###### Acknowledgements.

We thank the German Research Foundation (DFG) in the framework of the Research
Training Group “Functionalization of Semiconductors” (GRK 1782) for funding
this project and to HRZ Marburg, GOETHE-CSC Frankfurt, ZIH Dresden, and HLR
Stuttgart for providing the computational resources.

## DATA AVAILABILITY STATEMENT

The density functional theory calculations data are openly available in the
NOMAD repository (https://doi.org/10.17172/NOMAD/2023.02.27-1). The
interactive bandgap phase diagrams (in HTML format) are available in the
Supplemental Material [94]. To view the diagrams, open the HTML files in a web
browser. Alternatively, the diagrams can be viewed directly on GitHub
(https://bmondal94.github.io/Bandgap-Phase-Diagram/, last accessed
10.05.2023).

## ORCID iDs

Badal Mondal
https://orcid.org/0000-0002-0522-1254
Ralf Tonner-Zech
https://orcid.org/0000-0002-6759-8559
Thilo Hepp
https://orcid.org/0000-0002-0782-987X
Kerstin Volz
https://orcid.org/0000-0002-4456-5439

## Supplemental Material

The Supplemental Material contains the effective bandstructures of GaAsP,
GaAsN, and GaAsN under selected isotropic strain values; a visual
representation of the assessment of uncertainty in the bandgap nature near the
direct-indirect transition region; the DFT calculated bandgap values for all
the systems described in the manuscript; the details of the interpolation
procedure; and the bandgap phase diagrams.

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# A prime power equation

Timothy Redmond, Charles Ryavec

(Date: September 15, 2022)

###### Abstract.

A real valued function, $G$, is provided whose Fourier transform, $\hat{G}$,
is an entire function that satisfies,
$E(s)\zeta(s)=\hat{G}(\frac{s-\frac{1}{2}}{i})$. Then $\hat{G}(\gamma)=0$ for
all nonreal zeros, $\rho=\frac{1}{2}+i\gamma$, of $\zeta(s)$. Combined with
Guinand’s explicit formula we obtain a prime power equation free of zeta
zeros. Using infinitely many translates of G, an infinite system of equations,
indexed on the natural numbers, is obained. The solution vector of this system
is the vector of values of von Mangoldt’s function, $\Lambda(n),n=1,2\cdots$.
The entries of the matrix are special values of the fourth power of the Jacobi
theta function, $\theta_{2}(\tau)$.

##  1\. Explicit Formulas

The explicit formulas of prime number theory, as discovered by Riemann, and
generalized subsequently, provide a balance between two unknown sequences
within the framework of a Fourier transform pair, $G$ and $\hat{G}$. The
explicit formula considered in this note is that of Guinand:

$\displaystyle\sum_{\rho=\frac{1}{2}+i\gamma}\hat{G}(\gamma)$ $\displaystyle=$
$\displaystyle\hat{G}(-\frac{i}{2})+\hat{G}(\frac{i}{2})$ $\displaystyle=$
$\displaystyle-\sum_{p^{m}}\frac{logp}{p^{\frac{m}{2}}}\Big{(}G(m\ logp)+G(-m\
logp)\Big{)}$ $\displaystyle-$ $\displaystyle log\pi\ G(0)$ $\displaystyle+$
$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}Re\Big{(}\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}+\frac{it}{2})\Big{)}\hat{G}(t)dt,$

where the series on the left is the sum over nonreal zeros of the Riemann zeta
function, and in the series on the right, a sum over rational prime powers.

Various strategies have been brought to bear on this relationship in order to
gain results on zeros. The consideration of the left side,

$\sum_{\rho=\frac{1}{2}+i\gamma}\hat{G}(\gamma),$

over particular classes of G may be found in [1], [2], articles which provide
references and discussion. The first step here is to eliminate the sum,

$\sum_{\rho=\frac{1}{2}+i\gamma}\hat{G}(\gamma),$

altogether. To this end a function, $G$, is produced whose Fourier transform,
$\hat{G}$, is entire and factors as,

$\displaystyle E(s)\zeta(s)=\hat{G}(\frac{s-\frac{1}{2}}{i}),$

so that

$\displaystyle\hat{G}(\gamma)=0$

for all nonreal zeros, $\rho=\frac{1}{2}+i\gamma$, of $\zeta(s)$. (The
$\gamma$ are not assumed to be real.) In this case an explicit formula,

$\displaystyle\sum_{p^{m}}\frac{logp}{p^{\frac{m}{2}}}\Big{(}G(m\ logp)+G(-m\
logp)\Big{)}$ $\displaystyle=$ $\displaystyle V(G)$ $\displaystyle V(G)$
$\displaystyle=$ $\displaystyle\hat{G}(-\frac{i}{2})+\hat{G}(\frac{i}{2})$
$\displaystyle-$ $\displaystyle log\pi\ G(0)$ $\displaystyle+$
$\displaystyle\frac{1}{4\pi}\int_{-\infty}^{\infty}\Big{(}\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}+\frac{it}{2})+\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}-\frac{it}{2})\Big{)}\hat{G}(t)dt,$

exists, and a sum on prime powers is thereby expressed in terms of quantities
with no apparent connection with the zeros of the Riemann zeta function. It is
readily checked that the equation,

$\displaystyle\sum_{n=1}^{\infty}\frac{\Lambda(n)}{\sqrt{n}}\Big{(}G(log(x_{n})+G(-log(x_{n})\Big{)}=V(G),$

does not identify the prime powers; i. e., $x_{n}=n$. To pinpoint the prime
powers as a unique solution, it would be expected that infinitely many
equations would be needed. Given the one pair, $G(v)$ and $\hat{G}(t)$, the
obvious candidates for further $G$ are convolutions, the translations,
$G(v+c)$, being the simplest, with transforms $e^{-ict}\hat{G}(t)$. In a
coming section, we use the particular translates, $[c=logm:m=1,2,3,\cdots]$,
obtaining an infinite system of equations,

$\sum_{n=1}^{\infty}\Lambda(n)f(m,n)=V(m),$

indexed by the natural numbers, $[m=1,2,\cdots]$. The entries of the column
vector,

$V=[V(m)]_{m=1}^{\infty},$

are the values of the $m$-th translate, $V(G(v+logm))$. The objective would be
a unique solution vector,

$\Lambda=[\Lambda(n)]_{n=1}^{\infty},$

of

$T\Lambda=V,$ $T=[f(m,n)]_{m,n=1}^{\infty},$

with $\Lambda(n)$ being the von Mangoldt function. The entries, $f(m,n)$, (in
the $m$-th row and $n$-th column) of $T$ are essentially special values of the
modular function,

$\lambda(\tau)=\Big{(}\frac{\theta_{2}(\tau)}{\theta_{3}(\tau)}\Big{)}^{4}.$

These are numbers from finite algebraic extensions of the rational field, $Q$.
It was not shown that $T$ guaranteed a unique solution. Alternatively, instead
of creating equations from the translates of one $G$, a possible area to look
into for other $G$ are Galois extensions, $L:Q$, with

$E_{L}(s)\zeta_{L}(s)=\hat{G}_{L}(\frac{s-\frac{1}{2}}{i}),$

since $\zeta(s)$ is a factor of all such $\zeta_{L}$. We checked that this was
possible for 15 imaginary quadratic extensions of $Q$, and found that such
$\hat{G}_{L}(\frac{s-\frac{1}{2}}{i})$ existed. The new ingredient is a whole
new set of unrelated $\hat{G}_{L}$.

##  2\. A Zeta Function Representation

We define two multiplicative functions, $a(n)$ and $b(n)$.

######  Definition 1.

Define

$\displaystyle a(n)$ $\displaystyle=$
$\displaystyle\sigma_{1}(n),\qquad(n,2)=1$ $\displaystyle=$ $\displaystyle
2^{m}\qquad\qquad n=2^{m},$

and

$\displaystyle b[n]$ $\displaystyle=$
$\displaystyle\sigma_{1}(n)\qquad(n,2)=1$ $\displaystyle=$ $\displaystyle
0\qquad\ \ \ \ \ \ n=2^{m}\ \ \ m\geq 1.$

Note that differentiating the theta function,

$\displaystyle\theta_{4}(\tau)=\sum_{n=-\infty}^{\infty}(-1)^{n}q^{n^{2}}\qquad
q=e^{i\pi\tau},\tau=iy,$

with respect to y, has coefficients given by,

$\displaystyle\frac{\dot{\theta}_{4}}{\theta_{4}}(iy)$ $\displaystyle=$
$\displaystyle-2\pi\sum_{n=1}^{\infty}a(n)e^{-\pi ny},$

while the fourth power of the theta function, $\theta_{2}(\tau)$, has
coefficients,

$\displaystyle\theta_{2}^{4}(\tau)$ $\displaystyle=$
$\displaystyle\Big{(}\sum_{n=-\infty}^{\infty}q^{(n+\frac{1}{2})^{2}}\Big{)}^{4}$
$\displaystyle=$ $\displaystyle 16\sum_{n=1}^{\infty}b(n)q^{n}.$

######  Lemma 1.

$\displaystyle\frac{s(s+1)}{32\pi^{2}\sqrt{2}}(2^{\frac{s}{2}}-2^{-\frac{s}{2}})(2^{\frac{s-1}{2}}-2^{-\frac{s-1}{2}})\zeta^{*}(s)\zeta^{*}(s+1)$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{s}{2}}K(s,2\pi\sqrt{j}),$
$\displaystyle\zeta^{*}(s)$ $\displaystyle=$
$\displaystyle\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s).$

Here, $K(s,x)$ is the modified Bessel function of the second kind, usually
written, $K_{s}(x)$.

###### Proof.

As before, let

$\theta_{4}(\tau)=\sum_{k=-\infty}^{\infty}(-1)^{k}q^{k^{2}}\qquad
q=e^{i\pi\tau}.$

Then

$\theta_{4}^{4}(\tau)-1=-8\sum_{k=1}^{\infty}c(k)q^{k},$

with the multiplicative coefficients,

$\displaystyle c(k)$ $\displaystyle=$
$\displaystyle\sigma_{1}(k),\qquad(k,2)=1$ $\displaystyle=$
$\displaystyle-3\qquad\qquad k=2^{m},\ 1\leq m.$

With $\tau=iy$, and integrating $y$ over $(0,\infty)$, there results,

$\displaystyle\int_{0}^{\infty}y^{2w-1}(\theta_{4}^{4}(e^{-\pi y})-1)dy$
$\displaystyle=$
$\displaystyle-8\pi^{-2w}\Gamma(2w)\sum_{k=1}^{\infty}c(k)k^{-2w}$
$\displaystyle=$
$\displaystyle-8\pi^{-2w}\Gamma(2w)\sum_{(k,2)=1}\Big{(}1-3\sum_{m=1}^{\infty}2^{-2mw}\Big{)}c(k)k^{-2w}$
$\displaystyle=$
$\displaystyle-8\pi^{-2w}\Gamma(2w)(1-2^{2-2w})\zeta(2w-1)(1-2^{-(2w-1)})\zeta(2w)$
$\displaystyle=$
$\displaystyle-8\pi^{-(s+1)}\Gamma(s+1)(1-2^{1-s})\zeta(s)(1-2^{-s})\zeta(s+1),$

where, in the last line, we put $2w=s+1$. Returning to the integral in the
first line, and integrating by parts, we have,

$\displaystyle\int_{0}^{\infty}y^{2w-1}(\theta_{4}^{4}(e^{-\pi y})-1)dy$
$\displaystyle=$
$\displaystyle-\frac{4}{2w}\int_{0}^{\infty}y^{2w}\frac{\dot{\theta}_{4}}{\theta_{4}}(e^{-\pi
y})\theta_{4}^{4}(e^{-\pi y})dy$ $\displaystyle=$
$\displaystyle-\frac{4}{2w}\int_{0}^{\infty}y^{2w}\frac{\dot{\theta}_{4}}{\theta_{4}}(e^{-\pi
y})y^{-2}\theta_{2}^{4}(e^{-\frac{\pi}{y}})dy$ $\displaystyle=$
$\displaystyle\frac{4}{s+1}\int_{0}^{\infty}y^{s-1}2\pi\sum_{k=1}^{\infty}a(k)e^{-\pi
ky}(16)\sum_{l=1}^{\infty}b(l)e^{-\frac{\pi l}{y}}dy$ $\displaystyle=$
$\displaystyle\frac{4(2\pi)(16)}{s+1}\int_{0}^{\infty}y^{s-1}\sum_{k=1}^{\infty}a(k)e^{-\pi
ky}\sum_{l=1}^{\infty}b(l)e^{-\pi\frac{\pi l}{y}}dy$ $\displaystyle=$
$\displaystyle\frac{4(32)\pi}{s+1}\sum_{k=1}^{\infty}\sum_{l=1}^{\infty}a(k)b(l)\int_{0}^{\infty}y^{s-1}e^{-\pi(ky+\frac{l}{y})}dy$
$\displaystyle=$
$\displaystyle\frac{4(32)\pi)}{s+1}\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{s}{2}}\int_{0}^{\infty}y^{s-1}e^{-\pi(ky+\frac{l}{y})}dy$
$\displaystyle=$
$\displaystyle\frac{8(32)\pi}{s+1}\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{s}{2}}K(s,2\pi\sqrt{j}).$

Equating,

$-8\pi^{-(s+1)}\Gamma(s+1)(1-2^{1-s})\zeta(s)(1-2^{-s})\zeta(s+1)=\frac{8(32)\pi}{s+1}\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{s}{2}}K(s,2\pi\sqrt{j})$

and simplifying, the lemma follows. ∎

##  3\. $G(v)$ and $\hat{G}(t)$

With $K$ the modified Bessel function of the second kind, then, with
$s=\frac{1}{2}+it$, the Mellin transform is,

$\displaystyle K(\frac{1}{2}+it,x)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{0}^{\infty}u^{it}u^{-\frac{1}{2}}exp[-\frac{x}{2}(u+\frac{1}{u})]du.$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{-\infty}^{\infty}e^{itv}e^{\frac{v}{2}}exp[-\frac{x}{2}(e^{v}+e^{-v}))]dv$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{-\infty}^{\infty}e^{itv}g[v,x]dv$
$\displaystyle=$ $\displaystyle\frac{1}{2}\hat{g}(t,x),$

and the resulting pair of transforms collected here for future reference,

$\displaystyle g(v,x)$ $\displaystyle=$ $\displaystyle
e^{\frac{v}{2}}exp[-\frac{x}{2}(e^{v}+e^{-v}))]$ $\displaystyle\hat{g}(t,x)$
$\displaystyle=$ $\displaystyle\int_{0}^{\infty}e^{itv}g(v,x)dv$
$\displaystyle=$ $\displaystyle 2K(\frac{1}{2}+it,x).$

Note that if $\rho=\frac{1}{2}+i\gamma$, then

$\displaystyle\hat{g}(\gamma,x)=2K(\rho,x).$

Next, with $j$ a positive integer, and $d$ a positive integer divisor of $j$,
let

$\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{1}{2}Log\Big{(}\frac{j}{d^{2}}\Big{)}.$

Define

$\displaystyle G(v,d,j)$ $\displaystyle=$ $\displaystyle g(v-T,2\pi\sqrt{j})$
$\displaystyle=$ $\displaystyle
e^{\frac{v-T}{2}}exp[-\pi\sqrt{j}(e^{v-T}+e^{-(v-T)})]$ $\displaystyle=$
$\displaystyle\Big{(}\frac{j}{d^{2}}\Big{)}^{-\frac{1}{4}}e^{\frac{v}{2}}exp[-\pi\sqrt{j}\Big{(}e^{v}\frac{d}{\sqrt{j}}+e^{-v}\frac{\sqrt{j}}{d}\Big{)}]$
$\displaystyle=$
$\displaystyle\Big{(}\frac{j}{d^{2}}\Big{)}^{-\frac{1}{4}}e^{\frac{v}{2}}exp[-\pi\Big{(}e^{v}d+e^{-v}\frac{j}{d}\Big{)}].$

Then,

$\displaystyle\hat{G}[t,d,j]$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}e^{itv}g[v-T,2\pi\sqrt{j}]dv$
$\displaystyle=$ $\displaystyle e^{itT}\hat{g}[t,2\pi\sqrt{j}]$
$\displaystyle=$
$\displaystyle\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{it}{2}}\hat{g}[t,2\pi\sqrt{j}],$

and therefore,

$\displaystyle\hat{G}[\gamma,d,j]$ $\displaystyle=$
$\displaystyle\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{i\gamma}{2}}\hat{g}[\gamma,2\pi\sqrt{j}]$
$\displaystyle=$
$\displaystyle\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{i\gamma}{2}}2K(\rho,x).$

Finally, define,

######  Definition 2.

$\displaystyle G(v)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{1}{4}}G[v,d,j].$

Then,

$\displaystyle\hat{G}(t)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{1}{4}}\hat{G}[t,d,j]$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{1}{4}}\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{it}{2}}\hat{g}[t,2\pi\sqrt{j}]$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{\frac{1}{2}+it}{2}}\hat{g}[t,2\pi\sqrt{j}]$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{\frac{1}{2}+it}{2}}2K(\frac{1}{2}+it,2\pi\sqrt{j}]$

######  Theorem 1.

With G defined in Definition 2,

$\displaystyle\sum_{\gamma}\hat{G}(\gamma)$ $\displaystyle=$
$\displaystyle\sum_{\gamma}\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{1}{4}+i\frac{\gamma}{2}}2K(\frac{1}{2}+i\gamma,2\pi\sqrt{j})$
$\displaystyle=$ $\displaystyle 0.$

###### Proof.

$\displaystyle\hat{G}(\gamma)$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{1}{4}+i\frac{\gamma}{2}}2K(\frac{1}{2}+i\gamma,2\pi\sqrt{j})$
$\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{\rho}{2}}2K(\rho,2\pi\sqrt{j})$
$\displaystyle=$ $\displaystyle 0$

by Lemma 1. ∎

##  4\. The Prime Power Equation

######  Definition 3.

$\displaystyle\alpha(u)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}a(k)e^{-\pi ku}$ $\displaystyle\beta(u)$
$\displaystyle=$ $\displaystyle\sum_{l=1}^{\infty}b(l)e^{-\pi lu}\qquad u>0.$

With $G$ and $\hat{G}$ defined in section 3, the expressions in the prime
power equation,

$\displaystyle\sum_{p^{m}}\frac{logp}{p^{\frac{m}{2}}}\Big{(}G(m\ logp)+G(-m\
logp)\Big{)}$ $\displaystyle=$
$\displaystyle\hat{G}(-\frac{i}{2})+\hat{G}(\frac{i}{2})$ $\displaystyle-$
$\displaystyle log\pi\ G(0)$ $\displaystyle+$
$\displaystyle\frac{1}{4\pi}\int_{-\infty}^{\infty}\Big{(}\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}+\frac{it}{2})+\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}-\frac{it}{2})\Big{)}\hat{G}(t)dt,$

may be recast in terms of $\alpha$ and $\beta$, which is done in the next four
lemmas. Continuing,

######  Lemma 2.

$\displaystyle\sum_{p^{m}}\frac{logp}{p^{\frac{m}{2}}}\Big{(}G(m\ logp)+G(-m\
logp)\Big{)}=\sum_{n=1}^{\infty}\Lambda(n)\
\alpha(n)\beta(\frac{1}{n})+\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n}\alpha(\frac{1}{n})\beta(n).$

###### Proof.

Recalling,

$g[v,2\pi\sqrt{j}]=e^{\frac{v}{2}}exp[-\pi\sqrt{j}(e^{v}+e^{-v}))],$

then,

$\displaystyle G(mlogp,d,j)$ $\displaystyle=$ $\displaystyle g(m\
logp-T,2\pi\sqrt{j})$ $\displaystyle=$ $\displaystyle
e^{\frac{mlogp-T}{2}}exp[-\pi\sqrt{j}(e^{mlogp-T}+e^{-(mlogp-T)})]$
$\displaystyle=$ $\displaystyle
p^{\frac{m}{2}}\Big{(}\frac{d^{2}}{j}\Big{)}^{\frac{1}{4}}exp[-\pi\sqrt{j}\Big{(}p^{m}\frac{d}{\sqrt{j}}+p^{-m}\frac{\sqrt{j}}{d}\Big{)}]$
$\displaystyle=$ $\displaystyle
p^{\frac{m}{2}}\Big{(}\frac{d^{2}}{j}\Big{)}^{\frac{1}{4}}exp[-\pi\Big{(}p^{m}d+\
\frac{j}{p^{m}d}\Big{)}],$

so that

$\displaystyle G(m\ logp)$ $\displaystyle=$ $\displaystyle
p^{\frac{m}{2}}\sum_{j=1}^{\infty}\sum_{d|j}a[d]b[\frac{j}{d}]\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{1}{4}}\Big{(}\frac{d^{2}}{j}\Big{)}^{\frac{1}{4}}exp[-\pi\Big{(}p^{m}d+\
\frac{j}{p^{m}d}\Big{)}]$ $\displaystyle=$ $\displaystyle
p^{\frac{m}{2}}\sum_{j=1}^{\infty}\sum_{d|j}a[d]b[\frac{j}{d}]exp[-\pi\Big{(}p^{m}d+\
\frac{j}{p^{m}d}\Big{)}]$ $\displaystyle=$ $\displaystyle
p^{\frac{m}{2}}\sum_{k=1}^{\infty}a[k]exp[-\pi
p^{m}k]\sum_{l=1}^{\infty}b[l]exp[-\pi p^{-m}l]$ $\displaystyle=$
$\displaystyle p^{\frac{m}{2}}\alpha(p^{m})\beta(p^{-m}),$
$\displaystyle\sum_{p^{m}}\frac{logp}{p^{\frac{m}{2}}}G(m\ logp)$
$\displaystyle=$ $\displaystyle\sum_{p^{m}}logp\ \alpha(p^{m})\beta(p^{-m}),$

and

$\displaystyle G(-m\ logp,d,j)$ $\displaystyle=$ $\displaystyle g(-m\
logp-T,2\pi\sqrt{j})$ $\displaystyle=$ $\displaystyle
e^{\frac{-mlogp-T}{2}}exp[-\pi\sqrt{j}(e^{-mlogp-T}+e^{-(-mlogp-T)})]$
$\displaystyle=$ $\displaystyle
p^{-\frac{m}{2}}\Big{(}\frac{d^{2}}{j}\Big{)}^{\frac{1}{4}}exp[-\pi\sqrt{j}\Big{(}p^{-m}\frac{d}{\sqrt{j}}+p^{m}\frac{\sqrt{j}}{d}\Big{)}]$
$\displaystyle=$ $\displaystyle
p^{-\frac{m}{2}}\Big{(}\frac{d^{2}}{j}\Big{)}^{\frac{1}{4}}exp[-\pi\Big{(}p^{-m}d+\
p^{m}\frac{j}{d}\Big{)}],$
$\displaystyle\sum_{p^{m}}\frac{logp}{p^{\frac{m}{2}}}G(-mlogp)$
$\displaystyle=$
$\displaystyle\sum_{p^{m}}\frac{logp}{{p^{m}}}\sum_{k=1}^{\infty}a[k]exp[-\pi
p^{-m}k]\sum_{l=1}^{\infty}b[l]exp[-\pi p^{m}l]$ $\displaystyle=$
$\displaystyle\sum_{p^{m}}\frac{logp}{{p^{m}}}\alpha(p^{-m})\beta(p^{m}).$

This does Lemma 2. ∎

The rest of the terms on the right side of the prime power equation are
evaluated in Lemma 3, Lemma 4, and Lemma 5. Continuing with the term,
$-log\pi\ G(0)$, in the prime power equation,

######  Lemma 3.

$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a[d]b[\frac{j}{d}]\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{1}{4}}\Big{(}-log\pi\
G(0,d,j)\Big{)}$ $\displaystyle=$ $\displaystyle-log\pi\
\sum_{j=1}^{\infty}\sum_{d|j}a[d]b[\frac{j}{d}]e^{-\pi(d+\frac{j}{d})}$
$\displaystyle=$ $\displaystyle-log\pi\sum_{k=1}^{\infty}a[k]exp[-\pi
k]\sum_{l=1}^{\infty}b[l]exp[-\pi l]$ $\displaystyle=$ $\displaystyle-log\pi\
\alpha(1)\beta(1).$

######  Lemma 4.

$\displaystyle\hat{G}(\frac{i}{2})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a[d]b[\frac{j}{d}]\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{1}{4}}\hat{G}(\frac{i}{2},2\pi\sqrt{j})$
$\displaystyle=$ $\displaystyle
2\sum_{j=1}^{\infty}\sum_{d|j}a[d]b[\frac{j}{d}]K(0,2\pi\sqrt{j})$
$\displaystyle\hat{G}(-\frac{i}{2})$ $\displaystyle=$
$\displaystyle\sum_{j=1}^{\infty}\sum_{d|j}a[d]b[\frac{j}{d}]\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{1}{4}}\hat{G}(-\frac{i}{2},2\pi\sqrt{j})$
$\displaystyle=$ $\displaystyle
2\sum_{j=1}^{\infty}\sum_{d|j}a[d]b[\frac{j}{d}]\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{1}{2}}K(1,2\pi\sqrt{j}).$

######  Lemma 5.

$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}Re\Big{(}\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}+\frac{it}{2})\Big{)}\hat{G}(t)dt$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\frac{J(e^{v})}{1-e^{-2v}}dv-\gamma\alpha(1)\beta(1)\qquad\gamma=\mbox{Euler's
Constant}$ $\displaystyle J(e^{v})$ $\displaystyle=$ $\displaystyle
2e^{-2v}\alpha(1)\beta(1)-e^{-v}\alpha(e^{-v})\beta(e^{v})-\alpha(e^{v})\beta(e^{-v}).$

###### Proof.

We follow Bombieri ([1], pp. 188-190). From

$\displaystyle\Gamma[w]=\frac{1}{w}\prod_{n=1}^{\infty}\frac{(1+\frac{1}{n})^{w}}{1+\frac{w}{n}},$

there is

$\displaystyle\frac{\dot{\Gamma}}{\Gamma}(w)$ $\displaystyle=$
$\displaystyle-\frac{1}{w}+\sum_{n=1}^{N}\Big{(}log(1+\frac{1}{n})-\frac{1}{(n+w)}\Big{)}+\sum_{n>N}\Big{(}log(1+\frac{1}{n})-\frac{1}{(n+w)}\Big{)}$
$\displaystyle=$
$\displaystyle-\frac{1}{w}+logN-\sum_{n=1}^{N}\frac{1}{n+w}+\sum_{n>N}\Big{(}\frac{1}{n}-\frac{1}{n+w}\Big{)}+O\Big{(}\frac{1}{N}\Big{)}$
$\displaystyle=$ $\displaystyle
logN-\sum_{n=0}^{N}\frac{1}{n+w}+O\Big{(}\frac{1+|w|}{N}\Big{)}$

uniformly for $Re[w]\geq\frac{1}{4}$. Then

$\displaystyle
Re\Big{(}\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}+\frac{it}{2})\Big{)}=logN-\sum_{n=0}^{N}\frac{4n+1}{(2n+\frac{1}{2})^{2}+t^{2}}+O\Big{(}\frac{1+|t|}{N}\Big{)}.$

Thus,

$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}Re\Big{(}\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}+\frac{it}{2})\Big{)}\hat{G}(t)dt$
$\displaystyle=$ $\displaystyle
logN\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{G}(t)dt$ (1) $\displaystyle-$
$\displaystyle\sum_{n=0}^{N}\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{4n+1}{(2n+\frac{1}{2})^{2}+t^{2}}\hat{G}(t)dt$
$\displaystyle+$ $\displaystyle O(\frac{1}{N}).$

As for (1), the factor, $logN$, is written as,

$\displaystyle logN$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{N}\frac{1}{n}-\gamma+O(\frac{1}{N})$
$\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}e^{-2v}\Big{(}\frac{1-e^{-2vN}}{1-e^{-2v}}\Big{)}dv-\gamma+O(\frac{1}{N}).$

So (1) becomes,

$\displaystyle logN\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{G}(t)dt$
$\displaystyle=$ $\displaystyle
2\int_{0}^{\infty}e^{-2v}\Big{(}\frac{1-e^{-2vN}}{1-e^{-2v}}\Big{)}G(0)dv-\gamma
G(0)+O(\frac{1}{N})$ (3)

As for (2), write

$\displaystyle\frac{4n+1}{(2n+\frac{1}{2})^{2}+t^{2}}=\frac{1}{i}\Big{(}\frac{1}{t-i(2n+\frac{1}{2})}-\frac{1}{t+i(2n+\frac{1}{2})}\Big{)},$
(4)

and note that,

$\displaystyle\frac{1}{2\pi
i}\int_{-\infty}^{\infty}e^{itv}\Big{[}\frac{1}{t-i(2n+\frac{1}{2})}-\frac{1}{t+i(2n+\frac{1}{2})}\Big{]}dt=\text{Min}[e^{v(2n+1)},e^{-v(2n+1)}].$
(5)

Write (2) as a double integral,

$\displaystyle-\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{4n+1}{(2n+\frac{1}{2})^{2}+t^{2}}\hat{G}(t)dt=-\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{4n+1}{(2n+\frac{1}{2})^{2}+t^{2}}\int_{-\infty}^{\infty}e^{itv}G(v)dvdt,$

substitute (4), interchange the order of integration, and use (5) to obtain
for (2) the sums,

$\displaystyle-\sum_{n=0}^{N}\int_{-\infty}^{0}e^{v(2n+\frac{1}{2})}G(v)dv-\sum_{n=0}^{N}\int_{0}^{\infty}e^{-v(2n+\frac{1}{2})}G(v)dv,$

which equals,

$\displaystyle-\sum_{n=0}^{N}\int_{0}^{\infty}e^{v(2n+\frac{1}{2})}G(-v)dv-\sum_{n=0}^{N}\int_{0}^{\infty}e^{-v(2n+\frac{1}{2})}G(v)dv.$
(6)

Combining (3) and (6), and letting $N\longrightarrow\infty$, then,

$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}Re\Big{(}\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}+\frac{it}{2})\Big{)}\hat{G}(t)dt=$
$\displaystyle\int_{0}^{\infty}\Big{[}2e^{-2v}G(0)-e^{-\frac{v}{2}}(G(-v)+G(v))\Big{]}\frac{dv}{1-e^{-2v}}-\gamma
G(0).$

The expression in the square brackets is

$\displaystyle
J(e^{v})=2e^{-2v}\alpha(1)\beta(1)-e^{-v}\alpha(e^{-v})\beta(e^{v})-\alpha(e^{v})\beta(e^{-v}),$

so that, as stated in the lemma,

$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}Re\Big{(}\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}+\frac{it}{2})\Big{)}\hat{G}(t)dt=\int_{0}^{\infty}\frac{J(e^{v})}{1-e^{-2v}}dv-\gamma\alpha(1)\beta(1).$

Changing variables,

$\displaystyle v=logu\qquad dv=\frac{du}{u},$

we have,

$\displaystyle\int_{0}^{\infty}\frac{J(e^{v})}{1-e^{-2v}}dv$ $\displaystyle=$
$\displaystyle\int_{1}^{\infty}\frac{2u^{-2}\alpha(1)\beta(1)-u^{-1}\alpha(u^{-1})\beta(u)-\alpha(u)\beta(u^{-1})}{1-u^{-2}}\frac{du}{u}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{2u^{2}\alpha(1)\beta(1)-u\alpha(u)\beta(u^{-1})-\alpha(u^{-1})\beta(u)}{1-u^{2}}\frac{du}{u}$

∎

The term,

$\displaystyle V(\infty)$ $\displaystyle=$ $\displaystyle log\pi\ G(0)$
$\displaystyle+$
$\displaystyle\frac{1}{4\pi}\int_{-\infty}^{\infty}\Big{(}\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}+\frac{it}{2})+\frac{\dot{\Gamma}}{\Gamma}(\frac{1}{4}-\frac{it}{2})\Big{)}\hat{G}(t)dt,$

on the right side of the prime power equation comes from the Archimedian
factor in $Z(s)$, a component of the functional equation on which the Guinand
formula rests, and might as well be labelled that way. It plays the role of a
known term in the equations, but a difficult expression, containing the values
of $\beta(e^{-\pi y})$ in an integral. An effort was made to evaluate it as a
discrete expression of special values of $\beta$.

##  5\. Introducing a Variable $x>0$

Continuous translates, $G(v+logx),x>0,$ of $G$ are introduced, from which a
discrete infinite collection, $x=m,m=1,2,\cdots$ is selected to create an
infinite set of prime power equations. For $x>0$, it is readily verified that,

$\displaystyle
G(v+logx)=\sqrt{x}e^{\frac{v}{2}}\Big{(}\sum_{k=1}^{\infty}a(k)e^{-\pi
kxe^{v}}\Big{)}\Big{(}\sum_{l=1}^{\infty}b(l)e^{-\frac{\pi
l}{x}e^{-v}}\Big{)},$

so that

$\displaystyle G(mlogp+logx)$ $\displaystyle=$
$\displaystyle\sqrt{x}p^{\frac{m}{2}}\Big{(}\sum_{k=1}^{\infty}a(k)e^{-\pi
kxp^{m}}\Big{)}\Big{(}\sum_{l=1}^{\infty}b(l)e^{-\frac{\pi l}{xp^{m}}}\Big{)}$
$\displaystyle=$
$\displaystyle\sqrt{x}p^{\frac{m}{2}}\alpha(xp^{m})\beta(\frac{1}{xp^{m}})$
$\displaystyle G(-mlogp+logx)$ $\displaystyle=$
$\displaystyle\sqrt{x}p^{-\frac{m}{2}}\Big{(}\sum_{k=1}^{\infty}a(k)e^{-\pi
kxp^{-m}}\Big{)}\Big{(}\sum_{l=1}^{\infty}b(l)e^{-\frac{\pi
lp^{m}}{x}}\Big{)}$ $\displaystyle=$
$\displaystyle\sqrt{x}p^{-\frac{m}{2}}\alpha(\frac{x}{p^{m}})\beta(\frac{p^{m}}{x}).$

With the variable, $x$, in all the terms, the modified prime power equation is
now,

######  Lemma 6.

$\displaystyle\sqrt{x}\sum_{n=1}^{\infty}\Lambda(n)\
\alpha(xn)\beta(\frac{1}{xn})+\sqrt{x}\sum_{n=1}^{\infty}\frac{\Lambda(n)}{n}\alpha(\frac{x}{n})\beta(\frac{n}{x})$
$\displaystyle=$ $\displaystyle\frac{1}{\sqrt{x}}\hat{G}(\frac{i}{2})$
$\displaystyle+$ $\displaystyle\sqrt{x}\hat{G}(-\frac{i}{2})$ $\displaystyle-$
$\displaystyle\sqrt{x}\ log\pi\ \alpha(x)\beta(\frac{1}{x})$ $\displaystyle+$
$\displaystyle\sqrt{x}\int_{0}^{\infty}\frac{J(e^{v},x)}{1-e^{-2v}}dv-\gamma\sqrt{x}\
\alpha(x)\beta(\frac{1}{x}),$

where

$\displaystyle J(e^{v},x)$ $\displaystyle=$ $\displaystyle
2e^{-2v}\alpha(x)\beta(\frac{1}{x})-e^{-v}\alpha(xe^{-v})\beta(\frac{e^{v}}{x})-\alpha(xe^{v})\beta(\frac{e^{-v}}{x}).$

In order to remove $\hat{G}(\frac{i}{2})$ and $\hat{G}(-\frac{i}{2})$ from the
prime power equation, note that, given a relation,

$\displaystyle f(x)$ $\displaystyle=$ $\displaystyle\sqrt{x}g(x)$
$\displaystyle=$ $\displaystyle a\sqrt{x}+\frac{b}{\sqrt{x}}+h(x),$

as exists in our prime power formula, then the operation,

$f(x)+f(\frac{1}{x})-f(1)(\sqrt{x}+\frac{1}{\sqrt{x}})$

eliminates

$a\sqrt{x}+\frac{b}{\sqrt{x}}.$

This operation removes $\hat{G}[\frac{i}{2}]$ and $\hat{G}[-\frac{i}{2}]$. The
resulting prime power formula would depend only on $\alpha$ and $\beta$, but
as the situation stands, there is no evident advantage in taking this step,
and the right side, $V(G)$, is left with the two Bessel terms.

##  6\. Solving for Prime Powers

Writing the prime power equation as,

$\sum_{n=1}^{\infty}f(x,n)\Lambda(n)=V(x)\qquad x>0,$

and replacing $x$ with $m=1,2,\cdots$, in turn, there results a sequence of
equations,

$\sum_{n=1}^{\infty}f(m,n)\Lambda(n)=V(m),$

indexed by the natural numbers, $[m=1,2,\cdots]$. With the pair of column
vectors,

$\Lambda=[\Lambda(n)]_{n=1}^{\infty},$

and,

$V=[V(m)]_{m=1}^{\infty},$

the linear system of equations is put in the form,

$T\Lambda=V.$

The entries of

$T=[f(m,n)]_{m,n=1}^{\infty},$

are

$f(m,n)=\alpha(mn)\beta(\frac{1}{mn})+\frac{1}{n}\alpha(\frac{m}{n})\beta(\frac{n}{m}),$

where the square root, $\sqrt{m}$, was put into the $V$ vector. The objective
is a solution,

$\Lambda=T^{-1}V.$

From any angle this does not look promising.

##  7\. The Invertibility of $T$

Firstly, the possible inverse, $T^{-1}$, cannot be approached as a limit of
inverses of upper left hand $n\times n$ cutouts of $T$. These are not
approximations to T. The diagonal entries of $T$ are

$f(n,n)=\frac{\alpha(1)\beta(1)}{n}+\epsilon(n)\qquad\epsilon(n)\longrightarrow
0.$

Since $\alpha$ and $\beta$ are positive,

$\displaystyle\sum_{k=n+1}^{2n-1}f(n,k)$ $\displaystyle\geq$
$\displaystyle\sum_{k=n+1}^{2n}\frac{1}{k}\alpha\Big{(}\frac{n}{k+1}\Big{)}\beta\Big{(}\frac{k+1}{n}\Big{)}$
$\displaystyle\geq$ $\displaystyle log2\ \alpha(1)\beta(2).$

Thus $T$ is not diagonally dominant, in fact, as shown, far from it. And the
arguments used to establish the existence of an inverse for infinite systems,
which rely on finite dimensional (upper left) approximations to $T$, cannot be
applied.
Note that

$\displaystyle\sum_{k=n+1}^{2n-1}f(n,k)\Lambda(k)$ $\displaystyle\geq$
$\displaystyle\sum_{k=n+1}^{2n}\frac{\Lambda(k)}{k}\alpha\Big{(}\frac{n}{k+1}\Big{)}\beta\Big{(}\frac{k+1}{n}\Big{)}$
$\displaystyle\geq$ $\displaystyle log2\ \alpha(1)\beta(2),$

so the action of the $n$th row of the finite dimensional, $n\times n$, upper
left truncation, $T_{n}$, of $T$, on $\Lambda$ differs from the action of the
$n$th row of $T$ on $\Lambda$ by a constant for all $n$.

##  8\. The Matrix T

Secondly, assuming an inverse, $T^{-1}$, finding it contains the known
complications of finding an inverse in the finite dimensional case. Even
matrices with simple entries might have an inverse that relies on a highly
sophisticated combinatorial identity. Basically, they’re works of art.
Consider what there is to encounter in the entries of $T$. Since

$\displaystyle\alpha(u)=\sum_{l=0}^{\infty}2^{l}\beta(2^{l}u),$

the matrix, $T$, depends on $\beta$ alone, and consists of sums and products
of values,

$\beta(e^{-\pi r})\qquad r\in Q^{+},$

which are, to within a factor of $\frac{1}{16}$, equal to

$\theta_{2}^{4}(\tau)\qquad q=e^{i\pi\tau},\tau=ir.$

For $r=N$, a positive integer,

$\theta_{2}^{4}(N\tau)=\Big{(}\frac{\theta_{3}(N\tau)}{\theta_{3}(\tau)}\Big{)}^{4}\lambda(N\tau)\theta_{3}^{4}(\tau),$

and the factor,

$F(N\tau)=\Big{(}\frac{\theta_{3}(N\tau)}{\theta_{3}(\tau)}\Big{)}^{4}\lambda(N\tau),$

is algebraic at $\tau=i$. With $N=p$, a prime, this follows from three
results. First, that

$\frac{\theta_{3}(p\tau)}{\theta_{3}(\tau)}$

is an algebraic function of $\lambda(\tau)$; second, that $\lambda(p\tau)$ is
an algebraic function of $\lambda(\tau)$; and, third, that
$\lambda(i)=\frac{1}{2}$. The values of $F(Ni)$ are then algebraic functions
of $\lambda(i)$, being compositions indexed on the prime factors of $N$. For
rational $r=\frac{N}{M}>0$, $F(ri)$ is algebraic, arguing as before with
$\tau=\frac{i}{M}$, using the relation,

$\lambda(\frac{i}{M})=1-\lambda(iM).$

Finally,

$\displaystyle\theta_{3}(i)$ $\displaystyle=$
$\displaystyle\frac{\pi^{\frac{1}{4}}}{\Gamma(\frac{3}{4})}.$

Thus, the values, $\theta_{2}^{4}(ri)$, the building blocks of the entries of
$T$, are algebraic multiples of $\theta_{3}^{4}(i)$.

##  9\. Supposing $T^{-1}$

Write,

$\Lambda=T^{-1}V.$

Let

$U=[u_{mn}]_{m,n=1}^{\infty}$ $\displaystyle u_{mn}$ $\displaystyle=$
$\displaystyle 1\qquad n\leq m$ $\displaystyle=$ $\displaystyle 0\qquad n>m$
$\displaystyle=$ $\displaystyle\frac{1}{2}\qquad n=m=p^{k},$

then

$U\Lambda=\Psi_{0},$

where $\Psi_{0}$ is the column vector,

$[\Psi_{0}(N)]_{N=1}^{\infty},$

with

$\Psi_{0}(N)=\sum_{n\leq N}u_{Nn}\Lambda(n).$

Having reached this point, it would then seem that technique is within reach
to reconcile this formula with the explicit,

$\Psi_{0}(N)=N-\sum_{\rho}\frac{N^{\rho}}{\rho}-\frac{\dot{\zeta}}{\zeta}(0)-\frac{1}{2}log(1-N^{-2})\qquad
N>1,$

and identify the zeros of the zeta function originating in the structure of
$T$. The point here is that a mystery has simply been recast in a slightly
novel setting. None of the mystery has been eliminated. The unknown, $T^{-1}$,
was therefore an expected stopping point, though a $T^{-1}$ might be
investigated in the context of function fields, where the (norms of) primes
have an internal structure,

$p=q^{e(p)}\qquad e(p)\in Z^{+}.$

With

$e_{n}=\Big{|}[p:e(p)=n]\Big{|},$

the sequence,

$N_{k}=\sum_{d|k}de_{d}$

satisfies a finite linear recursion over the integers. Its characteristic
polynomial produces the zeros of a zeta function. If

$\displaystyle p$ $\displaystyle=$ $\displaystyle e^{logp}$ $\displaystyle=$
$\displaystyle\prod_{n=1}^{\infty}e^{T_{mn}^{-1}V(n)}\qquad m=m(p),$

is the internal structure in rational primes, then a connection over $m(p)$
might be discoverable. It could replace the finite recursion, and the zeta
zeros would come out of that connection.

##  10\. Other $G$, $\hat{G}$

As we saw in Lemma 1, a $G$ can be constructed from the modular form,
$\theta_{4}^{4}$, which provides the equation,

$\displaystyle\frac{s(s+1)}{32\pi^{2}\sqrt{2}}(2^{\frac{s}{2}}-2^{-\frac{s}{2}})(2^{\frac{s-1}{2}}-2^{-\frac{s-1}{2}})\zeta^{*}(s)\zeta^{*}(s+1)=\sum_{j=1}^{\infty}\sum_{d|j}a(d)b(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{s}{2}}K[s,2\pi\sqrt{j}].$

As the translates of $G$ created equations with the question of an inverse
unanswered (more general convolutions were no more helpful), the search turned
to $G$ by other constructions. One such $G$ comes from $\theta_{4}^{8}$,
which, by the same procedure as with $\theta_{4}^{4}$, gives

$\displaystyle\frac{s(s+2)(s+3)}{256\pi^{3}\sqrt{2}}2^{\frac{s}{2}}(2^{\frac{s-1}{2}}-2^{-\frac{s-1}{2}})\zeta^{*}(s)\zeta^{*}(s+3)=\sum_{j=1}^{\infty}\sum_{d|j}a(d)B(\frac{j}{d})\Big{(}\frac{j}{d^{2}}\Big{)}^{\frac{s}{2}}K[s,2\pi\sqrt{j}].$

This paper could have been written with a $G$ and $\hat{G}$ coming from
$\theta_{4}^{8}$. Then $\beta^{2}$ would replace $\beta$. Everything else,
including $\alpha$, would be the same. The only powers of $\theta_{4}$ that
work in this way, however, are $1,2,4,8$. A countable replacement for the $x$
variable that would provide an infinite set of equations was sought from the
imaginary quadratic extensions. There are infinitely many of these, and the
$\theta_{3}$ type modular forms, (two-dimensional) on which the zeta functions
are constructed, might be replaced with $\theta_{4}$ type modular forms that
support the steps that give,

$\displaystyle E_{L}(s)\zeta_{L}(s)=\hat{G}_{L}(\frac{s-\frac{1}{2}}{i}).$

For example, the form producing $\zeta_{L}(s)$ for

$\displaystyle L=Q(\sqrt{-1})$

is

$\displaystyle\sum_{m,n=-\infty}^{\infty}e^{i\pi\tau(m^{2}+n^{2})},$

but

$\displaystyle\sum_{m,n=-\infty}^{\infty}(-1)^{m}(-1)^{n}e^{i\pi\tau(m^{2}+n^{2})},$

also gives a $\zeta_{L}(s)$, with a factor $1-2^{1-s}$. The field,

$\displaystyle L=Q(\sqrt{-2})$

with the form,

$\displaystyle\sum_{m,n=-\infty}^{\infty}e^{i\pi\tau(m^{2}+2n^{2})},$

is the other field with $1-2^{1-s}$. The other seven class number one fields
have a factor,

$\displaystyle 1-4^{1-s},$

as do $Q(\sqrt{-5})$ and $Q(\sqrt{-6})$. These are a bit tricky, and whether
the good construction persists is unknown. Even if there are infinitely many
good quadratic fields, it’s another $T$ dragging unsolved problems from one
branch of number theory into the question of another $T^{-1}$. The opposite
consideration is there, though, that the structure within which the prime
power equation should be considered, if it should be considered at all, is
necessarily a great deal more polished than the structure within which it
appeared.

##  11\. Bibliography

1\. E. Bombieri, Remarks on Weil’s quadratic functional in the theory of prime
numbers, I. Rend. Mat. Acc. Lincei s. 9, v. 11:183-233 (2000)
2\. H. Yoshida, On Hermitian Forms attached to Zeta Functions. In: N. Kurokawa
- T. Sunada (eds.), Zeta Functions in Geometry. Advanced Studies in Pure
Mathematics, 21, Mathematical Society of Japan, Kinokuniya, Tokyo 1992, 281 -
325.