# Anomalous excitonic phase diagram in band-gap-tuned Ta2Ni(Se,S)5

Cheng Chen  These authors contributed equally to this work Department of
Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom  Department of
Applied Physics, Yale University, New Haven, Connecticut 06511, USA  Weichen
Tang  These authors contributed equally to this work Physics Department,
University of California, Berkeley, California 94720, USA  Materials Science
Division, Lawrence Berkeley National Lab, Berkeley, California 94720, USA
Xiang Chen  Physics Department, University of California, Berkeley, California
94720, USA  Materials Science Division, Lawrence Berkeley National Lab,
Berkeley, California 94720, USA  Zhibo Kang  Department of Applied Physics,
Yale University, New Haven, Connecticut 06511, USA  Shuhan Ding  Department of
Physics and Astronomy, Clemson University, Clemson, South Carolina 29631, USA
Kirsty Scott  Department of Physics, Yale University, New Haven, Connecticut
06511, USA  Siqi Wang  Department of Applied Physics, Yale University, New
Haven, Connecticut 06511, USA  Zhenglu Li  Physics Department, University of
California, Berkeley, California 94720, USA  Materials Science Division,
Lawrence Berkeley National Lab, Berkeley, California 94720, USA  Jacob P.C.
Ruff  Cornell High Energy Synchrotron Source, Cornell University, Ithaca, New
York 14853, USA  Makoto Hashimoto  Stanford Synchrotron Radiation Lightsource,
SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA  Dong-
Hui Lu  Stanford Synchrotron Radiation Lightsource, SLAC National Accelerator
Laboratory, Menlo Park, California 94025, USA  Chris Jozwiak  Advanced Light
Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Aaron Bostwick  Advanced Light Source, Lawrence Berkeley National Laboratory,
Berkeley, California 94720, USA  Eli Rotenberg  Advanced Light Source,
Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Eduardo H. da Silva Neto  Department of Physics, Yale University, New Haven,
Connecticut 06511, USA  Robert J. Birgeneau  Physics Department, University of
California, Berkeley, California 94720, USA  Materials Science Division,
Lawrence Berkeley National Lab, Berkeley, California 94720, USA  Department of
Materials Science and Engineering, University of California, Berkeley,
California 94720, USA  Yulin Chen  Department of Physics, University of
Oxford, Oxford, OX1 3PU, United Kingdom  Steven G. Louie<EMAIL_ADDRESS>Physics Department, University of California, Berkeley, California 94720, USA
Materials Science Division, Lawrence Berkeley National Lab, Berkeley,
California 94720, USA  Yao Wang<EMAIL_ADDRESS>Department of Physics and
Astronomy, Clemson University, Clemson, South Carolina 29631, USA  Department
of Chemistry, Emory University, Atlanta, GA 30322, USA  Yu He<EMAIL_ADDRESS>Department of Applied Physics, Yale University, New Haven, Connecticut 06511,
USA  Physics Department, University of California, Berkeley, California 94720,
USA  Materials Science Division, Lawrence Berkeley National Lab, Berkeley,
California 94720, USA  These authors contributed equally to this work
Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom
Department of Applied Physics, Yale University, New Haven, Connecticut 06511,
USA  These authors contributed equally to this work Physics Department,
University of California, Berkeley, California 94720, USA  Materials Science
Division, Lawrence Berkeley National Lab, Berkeley, California 94720, USA
Physics Department, University of California, Berkeley, California 94720, USA
Materials Science Division, Lawrence Berkeley National Lab, Berkeley,
California 94720, USA  Department of Applied Physics, Yale University, New
Haven, Connecticut 06511, USA  Department of Physics and Astronomy, Clemson
University, Clemson, South Carolina 29631, USA  Department of Applied Physics,
Yale University, New Haven, Connecticut 06511, USA  Department of Applied
Physics, Yale University, New Haven, Connecticut 06511, USA  Physics
Department, University of California, Berkeley, California 94720, USA
Materials Science Division, Lawrence Berkeley National Lab, Berkeley,
California 94720, USA  Cornell High Energy Synchrotron Source, Cornell
University, Ithaca, New York 14853, USA  Stanford Synchrotron Radiation
Lightsource, SLAC National Accelerator Laboratory, Menlo Park, California
94025, USA  Stanford Synchrotron Radiation Lightsource, SLAC National
Accelerator Laboratory, Menlo Park, California 94025, USA  Advanced Light
Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley,
California 94720, USA  Advanced Light Source, Lawrence Berkeley National
Laboratory, Berkeley, California 94720, USA  Department of Physics, Yale
University, New Haven, Connecticut 06511, USA  Physics Department, University
of California, Berkeley, California 94720, USA  Materials Science Division,
Lawrence Berkeley National Lab, Berkeley, California 94720, USA  Department of
Materials Science and Engineering, University of California, Berkeley,
California 94720, USA  Department of Physics, University of Oxford, Oxford,
OX1 3PU, United Kingdom<EMAIL_ADDRESS>Physics Department, University
of California, Berkeley, California 94720, USA  Materials Science Division,
Lawrence Berkeley National Lab, Berkeley, California 94720, USA  Department of
Materials Science and Engineering, University of California, Berkeley,
California 94720, USA<EMAIL_ADDRESS>Department of Physics and Astronomy,
Clemson University, Clemson, South Carolina 29631, USA  Department of
Chemistry, Emory University, Atlanta, GA 30322, USA<EMAIL_ADDRESS>Department
of Applied Physics, Yale University, New Haven, Connecticut 06511, USA
Physics Department, University of California, Berkeley, California 94720, USA
Materials Science Division, Lawrence Berkeley National Lab, Berkeley,
California 94720, USA

###### Abstract

During a band-gap-tuned semimetal-to-semiconductor transition, Coulomb
attraction between electrons and holes can cause spontaneously formed excitons
near the zero-band-gap point, or the Lifshitz transition point. This has
become an important route to realize bulk excitonic insulators – an insulating
ground state distinct from single-particle band insulators. How this route
manifests from weak to strong coupling is not clear. In this work, using
angle-resolved photoemission spectroscopy (ARPES) and high-resolution
synchrotron x-ray diffraction (XRD), we investigate the broken symmetry state
across the semimetal-to-semiconductor transition in a leading bulk excitonic
insulator candidate system Ta2Ni(Se,S)5. A broken symmetry phase is found to
be continuously suppressed from the semimetal side to the semiconductor side,
contradicting the anticipated maximal excitonic instability around the
Lifshitz transition. Bolstered by first-principles and model calculations, we
find strong interband electron-phonon coupling to play a crucial role in the
enhanced symmetry breaking on the semimetal side of the phase diagram. Our
results not only provide insight into the longstanding debate of the nature of
intertwined orders in Ta2NiSe5, but also establish a basis for exploring band-
gap-tuned structural and electronic instabilities in strongly coupled systems.

When the indirect band gap of a material is continuously reduced from positive
to negative, band theory predicts a semiconductor-to-semimetal Lifshitz
transition. However, such a transition may never be reached in the presence of
electron-hole Coulomb attraction. It is postulated that poor screening in low
carrier density semimetals and strong electron-hole binding in narrow gap
semiconductors can both result in the spontaneous formation of excitons [1,
2]. The condensation of excitons results in an insulating ground state, the
so-called ”excitonic insulator” (EI), which is separated from the normal state
by a dome-shaped phase boundary peaked around the Lifshitz point (see Fig. 1a)
[3]. This simple setup became a primary platform to search for condensed
excitons and other exotic correlated phases, as well as potential solid-state
BCS-BEC crossover phenomena [4, 5, 6, 7]. For example, adding electronic
anisotropy can further narrow the EI dome due to reduced Coulomb attraction
[8, 9], and sufficiently heavy holes may cause Wigner crystallization [10].
Moreover, band-gap-tuned BCS-BEC crossover has been recently suggested in 2D
moiré systems [11]. However, interband electron-phonon interaction can also
lead to a gapped charge-density-wave state that occurs concurrently with the
exciton condensation [4]. It remains unclear how the excitonic instability and
the density-wave instability interact with each other, and whether it is
possible at all to distinguish them. Since no super-transport (charge or heat)
is associated with the excitonic insulator phase, and its analogy to BCS
superconductivity is only formal [10], an effective experimental method to
distinguish the Coulomb and lattice channels is to identify a tuning parameter
that gives differentiating predictions of the phase diagram.

Figure 1: Band-gap-tuned semimetal-to-semiconductor transition in narrow band
gap systems. a Archetypal phase diagrams in indirect band gap controlled
excitonic insulator (EI) across the semimetal-to-semiconductor transition. b
Normal state electrical resistivity evolution with S-doping in Ta2Ni(Se,S)5. c
Structure motif of quasi-1D Ta2Ni(Se,S)5 system below and above transition
temperature $T_{s}$ (upper panel). The lattice distortion in the monoclinic
phase is exaggerated. Symbolic sketch of a minimal lattice model of the system
used to describe the low-energy interactions in the system (lower panel).
$\epsilon_{k}^{c}$ and $\epsilon_{k}^{v}$ are kinetic energies of conduction
and valence electrons. $\omega_{q}$ is the chain-shearing $B_{2g}$ phonon
frequency. $g$ is the interband electron-phonon coupling vertex. $V$ is the
electron-hole Coulomb interaction potential.

Band-gap-controlled materials that undergo semimetal-to-semiconductor
transitions provide an ideal platform to investigate this issue. However, the
experimental realization of such a transition in bulk systems has been
challenging. Divalent metals under hydrostatic pressure and uniaxial strain
were first proposed as promising candidates [12], but experimental evidence
was indirect and scarce without clear ways to separate Coulomb and electron-
phonon interaction effects. Recent experiments suggest a potential exciton
condensation in 1T-TiSe2 [13], where the band gap tunability is limited.
Quasi-one-dimensional (quasi-1D) ternary chalcogenide Ta2NiSe5 has recently
emerged as another leading EI candidate, exhibiting an orthorhombic-to-
monoclinic structural transition at $T_{s}=$ 329 K, concomitant with a
putative exciton condensation [14, 15]. The system consists of parallel
quasi-1D Ta and Ni chains (see Fig. 1c), where long-lived electron-hole pairs
are supposed to dwell on Ta 5$d_{x^{2}-y^{2}}$ and Ni 3$d_{xy}$ orbitals. On
the one hand, the sizable ground-state single-particle gap [16, 17, 18, 19],
minute structural distortion [14, 20], and a “dome-like” temperature-pressure
phase diagram [15, 10] are all ostensibly consistent with the predictions in a
band-gap-tuned EI. On the other hand, strong electron-phonon coupling effects
are also found to coexist with correlation effects [21, 22, 20, 23, 24, 25,
26, 27, 28, 29, 30, 31, 32]. With the presence of both interactions, the band
topology alone no longer sufficiently dictates the excitonic insulator phase
region, and it is unclear how the low-temperature broken symmetry state would
evolve across the Lifshitz transition point. In Ta2Ni(Se,S)5, this question
can be addressed since the value of the direct band gap can be effectively
tuned by the isovalent sulfur substitution of selenium [15, 33]. However, the
phase diagram of Ta2Ni(Se,S)5– especially where the broken symmetry phase ends
and where the normal state crosses from a gapless to a gapped electronic
structure with S-doping – has seen vastly contradicting results [14, 15, 27],
mainly due to the lack of direct probes of the single-particle band gap and
the order parameter of the broken symmetry. A general impact of S-substitution
on the low-energy electronic structure can be revealed through the normal-
state (defined here as the high-temperature symmetric structure phase)
resistivity measured along the chain direction (see Fig. 1b), which slowly
increases until an abrupt upturn around $70\%$ doping. Such an upturn
indicates a potential Lifshitz transition on the normal-state band structures.
Hence, S-doped Ta2NiSe5 constitutes an ideal platform to study band-gap-tuned
semimetal-to-semiconductor transition in an electron-phonon coupled correlated
system.

Figure 2: Structural order parameter of Ta2Ni(Se,S)5. a Change of the inter-
and intra-chain Ta-Ta bond angle $\beta-90^{\circ}$ as a function of
temperature and S-doping. b Top: experimental and DFT calculated
$\beta-90^{\circ}$ at zero temperature. Bottom: structural transition
temperature $T_{s}$ as a function of S-doping level, determined from XRD and
resistivity measurement. Figure 3: Electronic Lifshitz transition in
Ta2Ni(Se,S)5. ARPES spectra along $X-\Gamma-X$ direction for samples with
different S-doping levels. a High-temperature orthorhombic phase. b Low-
temperature monoclinic phase (except for 100% S-doping, where no structural
transition was evidenced down to 10 K). Spectra from linear horizontal (LH,
blue) and linear vertical (LV, red) polarization of incident light are
overlaid, highlighting Ta 5$d_{x^{2}-y^{2}}$ conduction band and Ni 3$d_{xy}$
valence bands respectively. VBT - valence band top. c DFT calculation of 22%
S-doped compound in high-temperature orthorhombic phase, and the evolution of
$E_{g}$ extracted from DFT calculation and photoemission spectra as a function
of S-doping. d DFT calculation of 22% S-doped compound in low-temperature
monoclinic phase, and the evolution of $2k_{F}$ extracted from DFT calculation
and photoemission spectra as a function of S-doping.

We first track the evolution of the broken symmetry phase boundary in
Ta2Ni(Se,S)5with direct measurements of the structural order parameter. In
bulk excitonic insulators, exciton condensation is always tied to a
simultaneous lattice distortion [10]. Employing high-resolution single-crystal
synchrotron XRD, Fig. 2a characterizes the temperature- and S-substitution-
dependent evolution of the lattice order parameter $\beta$ (also see
Supplementary Note 1), which is defined as the angle between the lattice $a$
and $c$ axes shown in Fig. 1c. We find the structural phase transition remains
second-order throughout the entire S-doping range. The ground-state order
parameter $\beta(T=0)$ monotonically decreases with increasing S-doping (Fig.
2b), and the structural transition temperature only approaches zero near full
S-substitution (Ta2NiS5) (bottom panel in Fig. 2b). This is notably different
from the electronic phase diagram inferred from resistivity (see Fig. 2b and
Supplementary Note 1), which instead suggests an abrupt collapse of the broken
symmetry phase at an intermediate S-doping level [15]. Therefore, it is
imperative to obtain a direct electronic structure view to determine the
normal state semimetal-to-semiconductor transition point $p_{L}$ along the
S-doping axis.

Figure 4: Band gap tuned phase diagram with electron-phonon coupling. a
Evolution of structural transition temperature $T_{s}$ and the maximum Ta atom
displacement $|X|$ as a function of S-doping in Ta2Ni(Se,S)5. The electronic
band structures within different regions are illustrated in the insets. b The
calculated average lattice displacement $|x_{\rm
ph}|=\sqrt{\langle\hat{x}_{\rm ph}^{2}\rangle}$ as a function of the band gap
$E_{g}$ at various electron-phonon coupling strengths. In the quantized phonon
model, the zero-point fluctuation gives a finite $|x_{\rm ph}|=1$ in the
decoupled limit. c Sketched phase diagrams of band-gap-tuned semimetal-to-
semiconductor transition in the strong (top) and weak (bottom) coupling
limits. Solid lines mark the phase boundaries, above which the shades show the
order fluctuations [20]. Red shade on the bands illustrates the electronic
states that participate in the phase transition. The horizontal dashed lines
denote $E_{F}$.

Previous studies have reported conflicting values of $p_{L}$ from resistivity,
Raman measurements [15, 27]. Here, we conduct high-resolution ARPES
measurement, which is a direct probe of electronic structure, to reveal the
evolution of band overlap/gap of Ta2Ni(Se,S)5 across the semimetal-
semiconductor phase transition. The high statistics of the measurements enable
electronic state restoration up to 4.5 kBT above EF (see Methods), and the
orbital selectivity through polarized X-ray beam, allows the direct comparison
with density functional theory (DFT) calculations, both of which help us
precisely locate $p_{L}$. Figure 3 shows the photoemission spectra in both the
normal and the broken symmetry state (also see Supplementary Note 1). As the
S-doping level increases, the Ta 5$d_{x^{2}-y^{2}}$ orbital component (blue)
gradually withdraws from the valence band top, which becomes dominated by the
Ni 3$d_{xy}$ orbital (red). Pseudogap states, the strong depletion of spectra
intensity around Fermi level [19, 20], are also observed, which account for
the high-temperature insulating behavior in these semimetallic samples (see
Ref.20 for detailed description). Beyond $p_{L}$ ($\sim 70\%$ S-doping), the
normal-state conduction and valence bands are completely separated in energy
(see Fig. 3a) and, simultaneously in the broken-symmetry-state, the Ta orbital
component largely disappears from the valence band top (see Fig. 3b). Fitting
the normal-state band dispersions (see Supplementary Note 2 for details)
quantifies the band gap (or band overlap in the case of negative values),
which changes sign across $p_{L}$ (Fig. 3c). In addition, revealed through the
“M-shape” dispersion of the valence band top in the broken symmetry state, the
band back-bending momentum $k_{F}$ also approaches $\Gamma$ near $p_{L}$ (Fig.
3d). Both results are consistent with a concurrent normal state semimetal-to-
semiconductor transition. Our DFT results further confirm such a Lifshitz
transition near $p_{L}$ via the band-gap-sign flip and the $k_{F}$ reduction
at the single-particle level, without involving ad hoc electron-hole Coulomb
attraction (see Fig. 3c-d and Supplementary Note 3).

The ARPES measurement of the semimetal-to-semiconductor transition doping is
consistent with that deduced from resistivity measurement (Fig. 1b and 2b).
However, the structural phase transition temperature $T_{s}$ and the
structural order parameter (approximated as $\beta_{\text{max}}$ \- 90∘ ) are
found to decrease monotonically with increasing S-doping through our XRD
measurements (Fig. 2b). This poses a marked departure from the archetypal
band-gap-tuned excitonic insulator phase diagram, where the excitonic
instability maximizes around $p_{L}$ (Fig. 1a). To highlight this contrast,
Figure 4a shows the evolution of both the measured structural transition
temperature $T_{s}$ and the experimentally determined Ta atom displacement
$|X|$ in Ta2Ni(Se,S)5. Both $|X|$ and $T_{s}$ appear insensitive to the system
crossing $p_{L}$, which leads to considerations beyond weak electron-hole
Coulomb interaction. One possibility is a heavy-hole Wigner crystal phase
[10]; but the electron and hole effective masses are both near the free
electron mass in Ta2Ni(Se,S)5 [33]. Alternatively, strong Coulomb interaction
may push the phase boundary towards the semiconductor side [34], but the
Hartree component will undo this effect[20] (see also Supplementary Note 4).
Considering the fact that strong electron-phonon coupling effect is revealed
in both equilibrium and ultrafast pump-probe experiments in Ta2Ni(Se,S)5 [21,
22, 20, 23, 24, 25, 26, 27, 28], we hereby investigate an experimentally
informed minimal model with strong electron-phonon coupling, shown in Fig. 1c
and Eq. (1) of Methods. This model contains both electron-hole Coulomb
interactions and an interband electron-phonon coupling, which can well
describe both the spectral properties of Ta2NiSe5 below and above $T_{s}$
[20]. To examine the impact of strong electron-phonon coupling on the band-
gap-tuned transition as a matter-of-principle question, we first set the
inter-band Coulomb interaction to zero. The role of this Coulomb interaction
primarily lies in a Hartree shift between two bands and can be mapped to a
correction of $E_{g}$ (see detailed discussions in Supplementary Note 4). The
average phonon displacement $|x_{\rm ph}|$ at $T=0$ is employed to approximate
the order parameters $|X|$ measured in the experiment (Fig. 4a). With the
quantized phonon modes, a zero-point fluctuation leads to the zero-coupling
baseline at $|x_{\rm ph}|=1$.

Solved with exact diagonalization, $|x_{\rm ph}|$ for various $E_{g}$ and
coupling strengths are shown in Fig. 4b. In the weak-coupling case (e.g.
$g=14$ meV), the lattice order parameter displays a dome-like structure
similar to the EI phase diagram derived based on Fermi-surface instability in
Fig. 1a. With increasing coupling strengths, the dome gradually evolves into a
monotonic crossover. This evolution exists not only for the many-body measure
of fluctuations, but also for order parameters when symmetry breaking is
allowed (see Supplementary Note 4). The monotonic crossover driven by strong,
non-perturbative interactions reflects the irrelevance of Fermi-surface
instability (top panel in Fig. 4c). The coupling strength in Ta2NiSe5,
estimated from both ARPES spectral fitting and first-principles deformation
potentials [20], is 50$\sim$60 meV, placing it in the strong-coupling regime.
While more complicated models with multiple bands may be necessary to
quantitatively reproduce the order parameter evolution, strong electron-phonon
coupling qualitatively describes the evolution of the broken-symmetry phase
across $p_{L}$ in experiments. Moreover, strong coupling also expands the
momentum range over which the lattice becomes unstable [35], opening up the
opportunity for a $\mathbf{q}=0$ transition when 2$k_{F}$ is small.

We demonstrate that strong electron-phonon interaction can greatly enhance the
structural symmetry breaking in a semimetal, and the evolution of the broken-
symmetry phase boundary can be indifferent to the semimetal-to-semiconductor
transition. On the specific debate over the nature of the symmetry breaking in
Ta2Ni(Se,S)5 systems, our results suggest a substantial deviation from the
electron-hole Coulomb attraction-based EI phase (bottom panel in Fig. 4c).
Instead, a strong electron-phonon-coupling induced structural transition can
explain the observed symmetry breaking. This conclusion is consistent with
recent experimental evidence of the crucial role of phonons in the equilibrium
and nonequilibrium properties of Ta2NiSe5 [22, 23]. Meanwhile, it has been
suggested recently that additional Coulomb and acoustic-phonon effects can
also cooperatively contribute to the transition [27]. On the high-temperature
fluctuations, previous investigations have identified a normal-state spectral
“pseudogap”, characterized by an anomalous spectral weight accumulation near
the conduction band bottom. This phenomenon is interpreted as the signature of
preformed excitons due to strong electronic correlation or electron coupling
to lattice fluctuations [36, 20]. We find that such a spectral anomaly
ubiquitously exists in the normal state on the semimetal side of Ta2Ni(Se,S)5,
but rapidly disappears on the semiconductor side (Fig. 3a). Intriguingly, the
boundary of the pseudogap, or preformed excitons, is almost temperature-
independent, similar to the abruptly terminated pseudogap in hole-doped
superconducting cuprates [37, 38]. Last but not least, the heavily
S-substituted Ta2Ni(Se,S)5 system provides a promising platform to study both
solid state BCS-BEC crossover phenomenon [39] and critical fluctuations of
low-energy phonons. Specifically, a potential competing order is observed in
Fig. 2a for S-dopings above $p_{L}$, similar to the situation in iron-based
high $T_{c}$ superconductors, where multiple orders are found competing with
each other at low temperatures, including nematicity, spin-density wave, and
superconductivity[40, 41].

## Methods

### Sample synthesis and resistivity measurements

Single crystals of Ta2Ni(Se,S)5 were grown by the chemical vapor transfer
method with iodine (I2) as the transport agent. Starting materials composed of
Ta powder (99.99%), Ni powder (99.99%), Se powder (99.99%), and S powder
(99.99%) with a nominal molar ratio 2:1:5(1-$x$):5$x$ ($x$ is the nominal
doping level of S) were fully ground and mixed together inside the glovebox.
About 50 mg of iodine (I2) was added to the mixture of the starting powder.
The mixture was then vacuumed, back-filled with 1/3 Argon, and sealed inside a
quartz tube with an inner diameter of 8 $mm$, an outer diameter of 12 $mm$,
and a length of about 120 $mm$. The sealed quartz tube was placed horizontally
inside a muffle furnace during the growth. The hot end reaction temperature
was set to 950 ∘C, and the cold end was left in the air with the temperature
stabilized at 850 ∘C. Long and thin single crystals were harvested by
quenching the furnace in the air after one week of reaction. Excess iodine was
removed from the surfaces of the crystals with ethanol. Electrical resistivity
measurements were carried out on a commercial PPMS (Quantum Design) by the
four-probe method with the current applied along the a-axis of the
Ta2Ni(Se,S)5 single crystals.

### Angle-resolved photoemission spectroscopy (ARPES)

Synchrotron-based ARPES measurements were performed at the beamline BL5-2 of
Stanford Synchrotron Radiation Laboratory (SSRL), SLAC, USA, and the BL 7.0.2
of the Advanced Light Source (ALS), USA. The samples were cleaved in situ and
measured under the ultra-high vacuum below 3×10-11 Torr. Data was collected by
the R4000 and DA30L analyzers. The total energy and angle resolutions were 10
meV and 0.2∘, respectively. High statistics measurements were done with a
signal-to-noise ratio of 100$\sim$150 at the Fermi level on the energy
distribution curve taken at the Fermi momentum. This enables electronic
structure restoration up to 4.5$\sim$5 $k_{B}T$ after Fermi function division
[42, 38]. Orbital content selection is based on the dipole transition matrix
element effect, following the experimental geometry described in Ref. [20].

### Single crystal x-ray scattering

Hard X-ray single-crystal diffraction is carried out at the energy of 44 keV
at the beamline QM2 of the Cornell High Energy Synchrotron Source (CHESS). The
needle-like sample is chosen with a typical lateral dimension of 100 microns,
then mounted with GE Varnish on a rotating pin before being placed in the
beam. A Pilatus 6M 2D area detector is used to collect the diffraction pattern
with the sample rotated 365∘ around three different axes at a 0.1∘ step and
0.1s/frame data rate at each temperature. The full 3D intensity cube is
stacked and indexed with the beamline software.

### First-principles DFT calculation

Ab initio calculations are performed using the Quantum ESPRESSO package [43,
44]. The crystal structure relaxations are performed using the r2SCAN
functional [45] with a semiempirical Grimme’s DFT-D2 van der Waals correction
[46]. S-doped compounds are simulated by the virtual crystal approximation
(VCA), where the pseudopotentials of Se and S atoms are linearly interpolated
(i.e. mixed together) according to the chemical composition. A
30$\times$30$\times$15 k-mesh and a 100 Ry wavefunction energy cut-off were
used. The electronic structure calculations are performed using the same
r2SCAN functional with the relaxed structures.

### Two-band model for many-body simulations

We consider the two-band model with inter-band e-ph couplings to describe the
physics in the Ta2Ni(Se,S)5 systems, whose Hamiltonian reads as [47]:

$\displaystyle\mathcal{H}(E_{g})$ $\displaystyle=$
$\displaystyle\sum_{k\sigma}\varepsilon_{k}^{c}(E_{g})c_{k\sigma}^{\dagger}c_{k\sigma}+\sum_{k\sigma}\varepsilon_{k}^{v}(E_{g})f_{k\sigma}^{\dagger}f_{k\sigma}+V\sum_{i,\sigma,\sigma^{\prime}}(n_{i\sigma}^{c}+n_{i+1\sigma}^{c})n_{i\sigma^{\prime}}^{f}$
(1)
$\displaystyle+\sum_{kq\sigma}\frac{g_{q}}{\sqrt{N}}\left[(a_{q}+a_{-q}^{\dagger})c_{k+q\sigma}^{\dagger}f_{k\sigma}+H.c.\right]+\sum_{q}\omega_{q}a_{q}^{\dagger}a_{q}$

where $c_{k\sigma}^{\dagger}$ ($c_{k\sigma}$) creates (annihilates) an
electron at the conduction band (primarily Ta 5$d$) for momentum k and spin
$\sigma$, with dispersion given by $\varepsilon_{k}^{c}$; and the
$f_{k\sigma}^{\dagger}$ ($f_{k\sigma}$) creates (annihilates) an electron at
the valence band (primarily Ni 3$d$), with dispersion given by
$\varepsilon_{k}^{v}$ (Fig. 1c). The $n_{i\sigma}^{c}$ and $n_{i\sigma}^{f}$
are the density operators for the conduction and valence band, respectively.
We employ the band structures fitted from the Ta2NiSe5 experiments, as
reported in Ref. 20, which determine the valence and conduction band
dispersion for a negative band gap $E_{g}=-0.3$ eV. Due to the difficulty of
fitting the full dispersions in the gapped Ta2Ni(Se,S)5 materials, we simplify
the model using a rigid band separation controlled by $E_{g}$. That being
said, the conduction and valence band structures for Ta2Ni(Se,S)5 read as

$\displaystyle\varepsilon^{c}_{k}(E_{g})=3.25-1.8\,{{\mathrm{cos}}}(k)-0.9\,{{\mathrm{cos}}}(2k)-0.6\,{{\mathrm{cos}}}(3k)+E_{g}/2$
(2)
$\displaystyle\varepsilon^{v}_{k}(E_{g})=-1.95+1.5\,{{\mathrm{cos}}}(k)+0.3\,{{\mathrm{cos}}}(2k)+0.1\,{{\mathrm{cos}}}(3k)-E_{g}/2\,.$
(3)

Here, $E_{g}=0$ indicates the Lifshitz transition where
$\varepsilon^{c}_{k=0}=\varepsilon^{v}_{k=0}$, while $E_{g}=0.3$ eV reflects
the situation of Ta2NiS5.

To approximate the lattice distortion in a finite-size simulation, we employ
the average displacement of phonons

$\displaystyle|x_{\rm ph}|=\sqrt{\mathrm{Tr}\left[\rho\hat{x}_{\rm
ph}^{2}\right]}\,,$ (4)

where $\rho$ is the density matrix of the e-ph system and $\hat{x}_{\rm
ph}=(a_{0}^{\dagger}+a_{0})/N$ is the uniform phonon displacement operator
[48].

## Acknowledgements

The authors wish to thank L. Kemper, E. Demler, D. Y. Qiu, P. J. Guo, P. A.
Volkov for helpful discussions. Use of the Stanford Synchrotron Radiation
Light Source, SLAC National Accelerator Laboratory, is supported by the US
Department of Energy, Office of Science, Office of Basic Energy Sciences under
Contract No. DE-AC02-76SF00515. This research used resources of the Advanced
Light Source, a US DOE Office of Science User Facility under Contract No. DE-
AC02-05CH11231. This work is based upon research conducted at the Center for
High Energy X-ray Sciences (CHEXS) which is supported by the National Science
Foundation under award DMR-1829070. Work at Lawrence Berkeley National
Laboratory was funded by the U.S. Department of Energy, Office of Science,
Office of Basic Energy Sciences, Materials Sciences and Engineering Division
under Contract No. DE-AC02-05-CH11231 within the Quantum Materials Program
(KC2202) which provided the numerical simulations and within the Theory of
Materials Program (KC2301) which provided the DFT calculations. S.D. and Y.W.
acknowledge support from U.S. Department of Energy, Office of Science, Basic
Energy Sciences, under Early Career Award No. DE-SC0022874. The electron-
phonon model and DFT simulations were performed on the Frontera and Stampede2
computing system, respectively, at the Texas Advanced Computing Center. The
work at Yale University is partially supported by the National Science
Foundation (NSF) under DMR-2239171.

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Supplementary Information
Anomalous excitonic phase diagram in band-gap-tuned Ta2Ni(Se,S)5

Cheng Chen

Weichen Tang

Xiang Chen

Zhibo Kang

Shuhan Ding

Kirsty Scott

Siqi Wang

Zhenglu Li

Jacob P.C. Ruff

Makoto Hashimoto

Dong-Hui Lu

Chris Jozwiak

Aaron Bostwick

Eli Rotenberg

Eduardo H. da Silva Neto

Robert J. Birgeneau

Yulin Chen

Steven G. Louie

Yao Wang

Yu He

##  I Supplementary Note 1: Complete data of resistivity, XRD, ARPES and EDX
measurements

Resistivity measurements were performed on each Ta2Ni(Se,S)5 sample with
different S-doping levels, and the result is illustrated in Fig. S1a. Similar
temperature evolution is evidenced as the sample becomes more insulating
towards low temperatures. However, the resistivity curve does not show clear
signs of metal-to-insulator transition. The structural symmetry-breaking phase
transition only manifests into a small kink, for instance around $T_{s}=$ 329
K in the pristine Ta2NiSe5. This feature can be better visualized as the peak
in activation energy $\Delta$ of the resistivity $\rho$, shown in Fig. S1b.
Such peaks are only evidenced in samples with S-doping levels below $72\%$
(result plotted in Fig. 2b of the main text). Therefore, we note that
transport measurement is not a direct measurement of the structural phase
transition.

Fig. S1: Resistivity measurement of Ta2Ni(Se,S)5. a Temperature-dependent
resistivity of Ta2Ni(Se,S)5. b The temperature dependence of the activation
energy $\Delta$ of the resistivity $\rho$ in a given by
$\Delta=-k_{B}T^{2}(\partial{\ln\rho}/\partial{T})$. The actual S doping level
$x$ of each sample is determined by energy-dispersive x-ray spectroscopy, in
which the typical variation of $x$ is less than 1$\%$.

Complete data for high-resolution synchrotron-based XRD measurement on
Ta2Ni(Se,S)5 family is illustrated in Fig. S2. Crossing the second-order
structural phase transition, the system turns from the high-temperature
orthorhombic phase to the low-temperature monoclinic phase, and some Bragg
peaks, for instance the 2 4 8 Bragg peak shown here, split into two. Such
splitting is evidenced in all samples with different S-doping levels, except
for Ta2NiS5 , where no structural transition is observed within the resolution
limit of the experiment. The temperature where the splitting starts marks the
structural phase transition point deduced from XRD measurement and the result
is summarized in Fig. 2a of the main text. On the other side, the evolution of
$\beta$ angle, i.e. structural order parameter in the low-temperature
monoclinic phase, can be deduced from the separation of the split peak, and
the result is plotted in Fig. 2b of the main text.

Fig. S2: High energy X-ray diffraction data of Ta2Ni(Se,S)5. Evolution of the
Bragg peak $\pm$ 2 4 8 as a function of temperature for samples with different
S-doping levels. The splitting of the peak marks the structural phase
transition from the high-temperature orthorhombic phase to the low-temperature
monoclinic phase. No structural transition is observed in Ta2NiS5 within the
resolution limit of the experiment (about 0.01 Å-1). Black dashed lines are
guides for the eyes of the Bragg peak positions.

High statistics and energy resolution ARPES measurements are performed on
Ta2Ni(Se,S)5 family at both the high-temperature orthorhombic phase and the
low-temperature monoclinic phase (except for at 100$\%$ S-doping where no
structural phase transition is observed) to trace the evolution of electronic
structure with S-doping. Since the structural phase transition is second order
in nature, we plot the structural order parameter at the temperature of ARPES
measurement in Fig. S3, to ensure that visible features of the monoclinic
phase have already developed. Linear horizontal (LH) and linear vertical (LV)
incident beams from synchrotron radiation are employed, selectively probing
bands with different orbital origins, which in this case highlight the
conduction and valence bands respectively. The complete data is illustrated in
Fig. S4. For the high-temperature data, we divide the resolution-convolved
Fermi-Dirac function to restore the spectra up to $\sim$150 meV above the
Fermi level. For the low-temperature data, as the samples become insulating
and the Fermi level is not pinned at a well-defined position, we aligned the
spectra of different samples to the top of the valence band to ensure a direct
comparison. For better visualization of the ARPES result, we combined the
intensity of LH and LV channels with equal contributions in blue and red
colors and plotted in Fig. 3 of the main text.

Fig. S3: Structural order parameter at the temperature of ARPES measurement
The amplitude of the deviation of the $\beta$ angle from $90^{\circ}$ at the
temperature of ARPES measurement, as compared to the maximum value. No
observable structural phase transition is observed at 100$\%$ S doped compound
(Ta2NiS5).  Fig. S4: Photoemission data of Ta2Ni(Se,S)5. a Photoemission
spectra along $X-\Gamma-X$ direction of Ta2Ni(Se,S)5 taken in high-temperature
orthorhombic phase. LH (linear horizontal) and LV (linear vertical) denote the
polarization of the incident photon beam. b Same as a but for spectra taken in
low-temperature monoclinic phase, except for the 100$\%$ sulfur substituted
compound (Ta2NiS5) where no sign of a structural phase transition is observed
in the XRD measurement.

To check the spatial homogeneity of sulfur doping within each doped sample, we
performed energy dispersive X-ray spectroscopy (EDX) mapping over the sample
surface, and Fig. S5 shows the results on the select samples with 64$\%$ and
72$\%$ S-doping levels. We find that the S-doping level within each sample is
homogeneous with an average variation of up to $\pm 2\%$ between regions of
10-20 microns lateral dimensions, which matches our smallest ARPES beam spot
sizes.

Fig. S5: Spatial homogeneity of S doping in Ta2Ni(Se,S)5. a(c) SEM image of
the surface of a sample with 64$\%$ and 72$\%$ S-doping. b(d) The false-color
plot of the ratio between the intensity of sulfur K-alpha peak and selenium L
peak corresponding to a(c). The green numbers on b(d) indicate the ratio
between the intensity of sulfur K-alpha peak and selenium L peak averaged in
the quarters segmented by the green dashed lines. The variation of the actual
S-doping level determined from our EDX measurements in each sample is
typically less than 1$\%$.

##  II Supplementary Note 2: Fitting of photoemission spectra

To estimate the band overlap/gap $E_{g}$ in the high-temperature ARPES
spectra, we fitted both the conduction and valence band dispersion. As
illustrated in Fig. S6, the spectra were first normalized along the energy
direction to compensate for the spectra weight depletion around the Fermi
level, resulting from the pseudogap state[20]. Then, the dispersion of the
bands was traced by peak-fitting the energy distribution curves (EDCs) or
momentum distribution curves (MDCs) of the ARPES spectra. The conduction band
is fitted with the parabolic curve (blue dashed line), and the valence band is
fitted using 3 different band shapes: linear (dashed yellow line), hyperbola
(red line), and DFT result (dashed orange line). The upper and lower bound of
$E_{g}$ comes from the fitting result of the linear and DFT band shapes,
respectively. In the case of quantifying low-temperature band back-bending
momentum $k_{F}$, similarly, we trace the dispersion of the valence band by
peak-fitting the EDCs and MDCs of the ARPES spectra. The 2$k_{F}$ value is
taken from the distance between the valence band top and the error comes from
the momentum resolution of the ARPES measurement.

Fig. S6: Photoemission spectra fitting. a High-temperature photoemission
spectra of a 22$\%$ S-doped sample in both LH (linear horizontal) and LV
(linear vertical) channels, highlighting conduction and valence band
respectively. The spectra are normalized along the energy axis to remove the
effect of the pseudogap state [20]. The “+” marks are deduced from the peak
fitting of energy distribution curves (EDCs) or momentum distribution curves
(MDCs), which were then fitted by hyperbola to extract the size of the band
overlap $E_{g}$. The upper (lower) bound of $E_{g}$ is estimated from the
fitting of the valence band with a linear (normalized DFT dispersion) band
shape. b Low-temperature photoemission spectra of the 22$\%$ S-doped sample in
LV channel. Band dispersion is deduced from the peak fitting of EDCs or MDCs,
from which the size of $2k_{F}$ is extracted.

##  III Supplementary Note 3: DFT calculation

Figure S7 shows complete data of DFT calculation on Ta2Ni(Se,S)5 in both the
orthorhombic and the monoclinic phases. The orthorhombic phase is achieved by
imposing the system’s symmetry to conform to the orthorhombic structure, while
the monoclinic phase is attained through full relaxation of the structure. The
result is overlayed on the ARPES spectra in Fig. 3a of the main text. The band
overlap/gap $E_{g}$ in the orthorhombic phase and the band back back-bending
momentum $k_{F}$ in the monoclinic phase are also deduced and the result is
plotted in Fig. 3c-d of the main text.

Fig. S7: DFT calculation of Ta2Ni(Se,S)5. Calculated band structure along
$X-\Gamma-X$ direction of Ta2Ni(Se,S)5 in a orthorhombic lattice structure, b
monoclinic lattice structure, respectively.

##  IV Supplementary Note 4: Impact of the Coulomb Interaction

To understand the impact of Coulomb interactions [the $V$ in Eq. (1) of the
main text], we incorporate non-zero interaction terms into the simulation of
the many-body system. Figure S8 shows the mean-average lattice displacement
$|x_{\rm ph}|$ evaluated using ED simulations for systems with $V=100$ meV and
200 meV. While considering the upper limit of the experimentally determined
Coulomb interaction around 70 meV[20], we allow for adjustments within a
certain range to accommodate potential variations induced by S-doping.
Compared to the Fig. 4 of the main text, the dependence of lattice
displacement on $E_{g}$ presented in Fig. S8 shows similar behavior with an
overall shift equal to the Coulomb interaction $V$. Such a shift can be better
characterized by the hump positions in the weak-coupling limit. The overall
move of the critical $E_{g}$ reflects a relative motion between two bands
caused by the Coulomb interaction $V$ and this phenomenon can be attributed to
the Hartree component.

Fig. S8: Influence of direct Coulomb interaction $V$. The calculated mean-
average lattice displacement $|x_{\rm ph}|$ as a function of the band gap
$E_{g}$ at various electron-phonon coupling strengths, with finite Coulomb
interaction (a) $V=100$ meV. and (b) $V=200$ meV. (The $V=0$ case is shown in
Fig. 4 of main text).  Fig. S9: Mean-field solutions for various interactions
and bare band gap sizes. Upper panels: Direct mean-field gap as a function of
electron-phonon coupling $g$ and bare band gap $E_{g}$, for interacting
electrons with $V=0$ (left), $V=0.05$ eV (middle), and $V=0.1$ eV (right).
Middle and lower panels: Mean-field order parameter for electrons $\Delta_{\rm
ext}$ (red) and lattice $X_{\rm ph}$ (blue).

In order to gain deeper insights into the influence of the parameter $V$, we
perform a mean-field (MF) analysis on the many-body Hamiltonian [Eq. (1) of
the main text]. Specifically, we consider the MF excitonic order parameter
$\Delta_{\rm ext}=\sum_{i\sigma}\langle
f^{\dagger}_{i\sigma}c_{i\sigma}\rangle/2N$ and $\Delta^{\prime}_{\rm
ext}=\sum_{i\sigma}\langle f^{\dagger}_{i\sigma}c_{i+1,\sigma}\rangle/2N$,
which are identical when the reflection symmetry is preserved, and the MF
lattice order parameter $X_{\rm ph}=\langle x_{\rm ph}\rangle$. The MF
decomposition of Eq. (1) is as follows

$\displaystyle\mathcal{H}_{\rm MF}$ $\displaystyle=$
$\displaystyle\sum_{k\sigma}(c_{k\sigma}^{\dagger},\,f_{k\sigma}^{\dagger})\left(\begin{array}[]{cc}\varepsilon^{c}_{k}+2V\sum_{i\sigma^{\prime}}\langle
n^{f}_{i\sigma^{\prime}}\rangle/N&gX_{\rm ph}-V(\Delta_{\rm ext}+{\rm
e}^{-{\rm i}k}\Delta^{\prime}_{\rm ext})\\\ gX_{\rm ph}-V(\Delta_{\rm
ext}+{\rm e}^{-{\rm i}k}\Delta^{\prime}_{\rm
ext})&\varepsilon^{v}_{k}+2V\sum_{i\sigma^{\prime}}\langle
n^{c}_{i\sigma^{\prime}}\rangle/N\end{array}\right)\left(\begin{array}[]{cc}c_{k\sigma}\\\
f_{k\sigma}\end{array}\right)$ (10)
$\displaystyle-\sum_{i\sigma\sigma^{\prime}}\langle
n^{c}_{i\sigma}\rangle\langle n^{f}_{i\sigma}\rangle+V(\Delta_{\rm
ext}^{2}+\Delta^{\prime\;2}_{\rm ext})+\frac{1}{2}N\omega_{0}X_{\rm ph}^{2},$

where the $V$ contributions to the diagonal terms reflect the Hartree shifts
between two bands, and $X_{\rm ph}$ is calculated by minimizing the ground-
state energy.

Solving the mean-field Hamiltonian yields the single-particle gap, which is
jointly determined by the Hartree shift and the hybridization arising from the
(off-diagonal) order parameters. As shown in the upper panels of Fig. S9, the
single-particle gap depends on both the e-ph coupling $g$ and the bard band
gap $E_{g}$. While a positive $E_{g}$, reflecting the S-doping in
Ta2Ni(Se,S)5, enhances the gap, this resultant gap opening is distinct from
the hybridization gap induced by symmetry breaking. This distinction can be
reflected in the lower panels, where the order parameters $\Delta_{\rm ext}$
and $X_{\rm ph}$ are strongly suppressed by positive $E_{g}$. Due to this
suppression, these order parameters display a dome-like structure centered
around $E_{g}=0$ in the case of a weak-coupling system. When the e-ph coupling
is strong, however, the hybridization gap is beyond the influence of $E_{g}$,
resulting in a monotonic reduction of order parameters as $E_{g}$ increases.
This behavior is consistent with the ED simulation depicted in the Fig. 4 of
the main text.

When incorporating the Coulomb interaction $V$ into this MF analysis, its
contribution to the order parameters is found to be minor, reflected by the
slight lowering of iso-intensity curves when contrasting the middle and left
panels. Instead, its primary contribution lies in uniformly shifting the
complete phase diagram towards smaller $E_{g}$ values. Therefore, the
Ta2Ni(Se,S)5 transition is primarily dictated by the strong e-ph coupling, as
opposed to the electronic Coulomb interactions.

The inefficiency of the Coulomb interaction in forming excitons results from
the small Fermi momentum and the opposite parity of the low-energy bands,
which has been discovered by earlier experimental and theoretical studies [21,
23]. The mismatch of parity completely excludes the purely Coulomb-driven
excitonic orders at $k=0$. This explains the diminishment of order parameters
for $E_{g}$, reflecting the reality close to the Ta2NiS5 side of the material.
While such an exclusion is no longer exact for finite Fermi momenta, the small
$k_{F}$ in Ta2NiSe5 leads to limited excitonic instability, compared to the
substantial Hartree shift caused by the Coulomb interaction [20]. In contrast,
the intraband $B_{2g}$ phonon connects the two bands directly and breaks the
mirror symmetry. Thus, the coupling to lattice distortion exhibits more
efficiency in forming excitons.


# Sums of Powers of Primes in Arithmetic Progression

Muhammet Boran  Department of Mathematics, Yıldız Technical University, 34220
Esenler, Istanbul, TURKEY<EMAIL_ADDRESS>,  John Byun
Department of Mathematics and Statistics, Carleton College, One North College
Street, Northfield, MN 55057, USA<EMAIL_ADDRESS>,  Zhangze Li
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
<EMAIL_ADDRESS>,  Steven J. Miller  Department of Mathematics and
Statistics, Williams College, Williamstown, MA 01267, USA<EMAIL_ADDRESS>and  Stephanie Reyes  Institute of Mathematical Sciences, Claremont Graduate
University, Claremont, CA 91711, USA<EMAIL_ADDRESS>
###### Abstract.

Gerard and Washington proved that, for $k>-1$, the number of primes less than
$x^{k+1}$ can be well approximated by summing the $k$-th powers of all primes
up to $x$. We extend this result to primes in arithmetic progressions: we
prove that the number of primes $p\equiv n\pmod{m}$ less than $x^{k+1}$ is
asymptotic to the sum of $k$-th powers of all primes $p\equiv n\pmod{m}$ up to
$x$. We prove that the prime power sum approximation tends to be an
underestimate for positive $k$ and an overestimate for negative $k$, and
quantify for different values of $k$ how well the approximation works for $x$
between $10^{4}$ and $10^{8}.$

###### Key words and phrases:

Prime Number Theorem, Arithmetic Progression

###### 2010 Mathematics Subject Classification:

Primary 11N13, Secondary 11N05.

This work was done in the 2023 Polymath Jr. program and was partially
supported by NSF award DMS-2113535.

##  1\. Introduction

###  1.1. Historical Context

Approximating the distribution of prime numbers is a major motivating question
in number theory. This task can be broken down into counting primes up to a
given number and estimating where prime numbers appear within the set of
natural numbers. The latter is a much more difficult task and falls outside of
the scope and context of our results. We instead extend a recent approximation
for the prime counting function to primes in arithmetic progressions. We use
analogous methods to the work of Gerard and Washington [GW], incorporating the
findings from Page [P] and Siegel [Si] and applying the prime number theorem
for arithmetic progressions where Gerard and Washington used the prime number
theorem. While our methods are similar, extending them to primes in arithmetic
progressions allows to investigate finer questions on the distributions of
prime numbers, for example, by studying Chebyshev bias.

######  Definition 1.1.

For $k\in\mathbb{R}$, define

$\pi(x)\ :=\ \sum\limits_{p\leq x}1\qquad\text{and}\qquad\pi_{k}(x)\ :=\
\sum\limits_{p\leq x}p^{k},$

where $p$ denotes a prime number here and throughout; $\pi(x)$ is the _prime-
counting function_.

######  Definition 1.2.

For real $x\geq 0$, define the _logarithmic integral_

$\operatorname{li}(x)\ :=\ \int_{0}^{x}\frac{dt}{\log t}.$

######  Definition 1.3.

Let $f$ and $g$ be two functions defined on $\mathbb{R}_{>0}$, and $g(x)$ be
strictly positive for all large enough values of $x$. We say

$f(x)\ =\ O(g(x))\qquad\text{as}\qquad x\to\infty$

if there exist constants $M$ and $x_{0}$ such that

$|f(x)|\ \leq\ M|g(x)|\qquad\text{for all}\qquad x\ \geq\ x_{0}.$

We also say

$f(x)\ll g(x)\qquad\text{is equivalent to}\qquad f(x)=O(g(x)).$

######  Definition 1.4.

Let $f$ and $g$ be two functions defined on $\mathbb{R}_{>0}$, and $g(x)$ be
strictly positive for all large enough values of $x$. We say

$f(x)\ =\ o(g(x))\qquad\text{as}\qquad x\to\infty$

if for every positive constant $\varepsilon$, there exists a constant $x_{0}$
such that

$|f(x)|\ \leq\ \varepsilon g(x)\qquad\text{for all}\qquad x\ \geq\ x_{0}$

and, equivalently,

$\lim_{x\to\infty}\frac{f(x)}{g(x)}\ =\ 0.$

We also say

$f(x)\lll g(x)\qquad\text{is equivalent to}\qquad f(x)=o(g(x)).$

######  Definition 1.5.

Given two functions $f$ and $g$, we say

$f(x)\sim g(x)\qquad\text{as}\qquad x\to\infty,$

read $f(x)$ _is asymptotic to_ $g(x)$, if

$\lim_{x\to\infty}\frac{f(x)}{g(x)}\ =\ 1$

and, equivalently, that

$f(x)\ =\ g(x)+O(h(x)).$

In 1896, Vallée-Poussin [V-P] proved that

$\pi(x)\sim\operatorname{li}(x)\qquad\text{as}\qquad x\to\infty,$

which is known as the prime number theorem, using techniques which relied on
complex analysis and the Riemann $\zeta$-function. Vallée-Poussin [V-P] also
estimated the error term in the prime number theorem by proving that

$\pi(x)\ =\ \operatorname{li}(x)+O\left(x\exp{\left(-c\sqrt{\log
x}\right)}\right)\ \ (x\geq 2)$ (1.1)

where $c$ is a positive constant. Further improvements on the error term began
to emerge. In 1922, Littlewood (outlined in [Ku]) proved that

$\pi(x)-\operatorname{li}(x)\ll x\exp{\left(-c\sqrt{\log x\log\log x}\right)}\
\ (x\geq 2).$

In 1958, Korobov [Ko] and Vinogradov [V] proved that

$\pi(x)-\operatorname{li}(x)\ll x\exp{\left(-c(\log x)^{3/5}(\log\log
x)^{-1/5}\right)}\ \ (x\geq 2).$

Since then, various alternative proofs have been given which do not use
complex analysis, with perhaps the most renowned being the proof by Erdős and
Selberg [Se].

A natural immediate extension of approximating the number of primes up to a
certain real number is estimating the _sum of primes_ up to a certain real
number. Using Vallée-Poussin’s result [Equation (1.1)], Szalay [Sz, Lemma 1]
found the following approximation for the sum of primes less than a real
number $x\geq 0$:

$\pi_{1}(x)\ =\
\operatorname{li}\left(x^{2}\right)+O\left(x^{2}\exp{\left(-c\sqrt{\log
x}\right)}\right).$ (1.2)

In 1996, Massias and Robin [MR, p. 217] approximated the sum of prime numbers
less than a real number $x\geq 0$ as

$\pi_{1}(x)\ =\
\operatorname{li}\left(\operatorname{li}^{-1}(x)^{2}\right)+O\left(x^{2}\exp{\left(-\gamma\sqrt{\log
x}\right)}\right)$

where $\gamma$ is a positive constant and $\operatorname{li}^{-1}(x)$ denotes
the inverse of $\operatorname{li}(x)$.

Massias and Robin [MR, Théorème D(v)] further bounded $\pi_{1}(x)$ above for
every $x\geq 24281$ by

$\pi_{1}(x)\ \leq\ \frac{x^{2}}{2\log x}+\frac{3x^{2}}{10\log^{2}x}.$ (1.3)

Around three decades following the publication of Szalay’s result [Equation
(1.2)], Axler [Ax] applied it to show that

$\pi_{1}(x)\ =\ \frac{x^{2}}{2\log
x}+\frac{x^{2}}{4\log^{2}x}+\frac{x^{2}}{4\log^{3}x}+\frac{3x^{2}}{8\log^{4}x}+O\left(\frac{x^{2}}{\log^{5}x}\right).$

Axler [Ax, Theorem 1.1] also improved on the upper bound given by Massias and
Robin [Line (1.3)] to give the following inequality

$\pi_{1}(x)\ <\ \frac{x^{2}}{2\log
x}+\frac{x^{2}}{4\log^{2}x}+\frac{x^{2}}{4\log^{3}x}+\frac{5.3x^{2}}{8\log^{4}x},$

which holds for every $x\geq 110118925$. For $x\geq 905238547$, Axler [Ax,
Theorem 1.2] also gives a lower bound of $\pi_{1}(x)$, which differs from the
upper bound only by a constant in the fourth term from the upper bound.

$\pi_{1}(x)\ >\ \frac{x^{2}}{2\log
x}+\frac{x^{2}}{4\log^{2}x}+\frac{x^{2}}{4\log^{3}x}+\frac{1.2x^{2}}{8\log^{4}x}.$

In 2016, shortly following Axler’s preprint, Gerard and Washington [GW,
Theorem 1] generalized these results to sums of fixed powers $k$ of primes.
They proved that $\pi_{k}(x)$ is asymptotic to $\pi(x^{k+1})$:

$\displaystyle\pi_{k}$ $\displaystyle(x)-\pi(x^{k+1})$
$\displaystyle=\begin{cases}O\left(x^{k+1}\exp\left(-A(\log x)^{3/5}(\log\log
x)^{-1/5}\right)\right)&\text{ if }k>0,\\\
O\left(x^{k+1}\exp\left(-(k+1)^{3/5}A(\log x)^{3/5}(\log\log
x)^{-1/5}\right)\right)&\text{ if }-1<k<0,\end{cases}$

where $A=.2098$. Their result is significant both from the perspective of
studying the prime counting function and from the perspective of studying sums
of powers of primes, as it relates the two functions with small relative error
for large $x$. From the perspective of studying the prime counting function,
their result allows for approximating $\pi(x)$ by
$\pi_{k}\left(x^{1/(k+1)}\right)$, which can significantly reduce the number
of primes used in the approximation.

###  1.2. Statement of Main Results

Throughout this paper, we take $m,n\in\mathbb{Z}_{>0}$ to be two coprime
positive integers, specifically with $n$ a unit in $\mathbb{Z}/m\mathbb{Z}$.
As mentioned previously, we use $p$ to denote prime numbers throughout.

To extend previous results, we first define the sum of $k$-powers of primes
for arithmetic progressions.

######  Definition 1.6.

For a fixed $k\in\mathbb{R}$, define

$\pi(x;m,n)\ =\ \sum\limits_{\begin{subarray}{c}p\leq x\\\ p\equiv
n\pmod{m}\end{subarray}}1\qquad\text{and}\qquad\pi_{k}(x;m,n)\ =\
\sum\limits_{\begin{subarray}{c}p\leq x\\\ p\equiv
n\pmod{m}\end{subarray}}p^{k}.$

Also, recall the definition of Euler’s totient function:

$\varphi(m)\ =\ \\#\\{x\in\mathbb{N}\ :\ x<m\ \text{and}\ \gcd(x,m)=1\\},$

which counts the number of positive integers up to $m$ which are coprime to
$m$.

We extend the work of Gerard and Washington [GW], studying sums of powers of
primes _in arithmetic progressions_. Dirichlet’s theorem on arithmetic
progressions states that there are infinitely many primes $p$ in the form
$p\equiv n\ (\mathrm{mod}\ m)$. Thus, $\pi(x;m,n)$ grows to $\infty$ as $x$
approaches $\infty$. Also, Vallée-Poussin [V-P] proved the _prime number
theorem on arithmetic progressions_ , concluding that

$\pi(x;m,n)\ \sim\ \frac{\pi(x)}{\varphi(m)}\ \sim\ \frac{x}{\varphi(m)\log
x}\qquad\text{as}\qquad x\to\infty,$ (1.4)

where $\gcd(m,n)=1$. A proof of (1.4) is outlined in [Ap, p. 154-155]. In
1935, Page [P] proved that there exists a positive constant $\delta$ such that

$\pi(x;m,n)\ =\
\frac{\operatorname{li}(x)}{\varphi(m)}+O\left(x\exp{\left(-(\log
x)^{\delta}\right)}\right)\ \ (x\geq 2),$

whenever $1\leq m\leq(\log x)^{2-\delta}$. In the same year, Siegel [Si]
proved that there exists a positive constant $\xi_{\eta}$ depending on $\eta$
such that

$\pi(x;m,n)\ =\
\frac{\operatorname{li}(x)}{\varphi(m)}+O\left(x\exp{\left(-\frac{\xi_{\eta}}{2}\sqrt{\log
x}\right)}\right)\ \ (x\geq 2),$

whenever $1\leq m\leq(\log x)^{\eta}$. We use the above version of the prime
number theorem on arithmetic progressions to obtain our main result.

Following the methods of Gerard and Washington in [GW] we obtain the following
theorem.

######  Theorem 1.1.

Fix a real number $k>-1$ and positive integers $m,n\in\mathbb{Z}_{>0}$ such
that $\gcd(m,n)=1$. Then we can approximate the number of primes $p\equiv
n\pmod{m}$ less than $x^{k+1}$ by the sum of $k$-powers of primes $p\equiv
n\pmod{m}$ less than a real number $x$:

$\displaystyle\pi_{k}$ $\displaystyle(x;m,n)-\pi(x^{k+1};m,n)$
$\displaystyle=\begin{cases}O\left(x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{\log
x}\right)\right)&\text{ if }k>0,\\\
O\left(x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{(k+1)\log
x}\right)\right)&\text{ if }-1<k<0,\end{cases}$

where $\alpha$ is a positive constant.

Additionally, generalizing the proof of [W, Theorem 2], we prove the
following.

######  Theorem 1.2.

Fix a real number $k>-1$ and positive integers $m,n\in\mathbb{Z}_{>0}$ such
that $\gcd(m,n)=1$. Then

$\int_{1}^{\infty}\frac{\pi_{k}(t;m,n)-\pi(t^{k+1};m,n)}{t^{k+2}}\,dt\ =\
-\frac{\log(k+1)}{(k+1)\varphi(m)}.$

Further,

$\displaystyle-\frac{\log(k+1)}{(k+1)\varphi(m)}\ $ $\displaystyle<\
0\qquad(k>0)$ $\displaystyle-\frac{\log(k+1)}{(k+1)\varphi(m)}\ $
$\displaystyle>\ 0\qquad(-1<k<0).$

######  Remark 1.

In 1853, Chebyshev [C] noticed that there are more primes of the form $p\equiv
3\pmod{4}$ than $p\equiv 1\pmod{4}$. Over the years the synonymous expression
“prime number races” has emerged to describe problems originated from
Chebyshev. Rubinstein and Sarnak [RS] provided a framework for the
quantification of Chebyshev’s bias in prime number races in arithmetic
progressions. They considered the sets $P(m;n_{1},n_{2})=\\{x\geq
2:\pi(x;m,n_{1})>\pi(x;m,n_{2})\\}$. With an appropriate notion of size, under
standard assumptions, they proved that $P(4;3,1)\ >\ P(4;1,3)$.

In Appendix 3.1, we provide calculations of various examples for different
values of $k$, $m$, and $n$. The integral computation in Theorem 1.2 confirms
the observations from our data that

$\pi_{k}(x;m,n)-\pi(x^{k+1};m,n)$

tends to be negative when $k>0$, and tends to be positive when $-1<k<0$ for
large values of $x$. Note that this does not definitively specify the sign of
the error for given large values of $x$. Since our result integrates over all
reals greater than one, we can only conclude that the “net” sign will either
be negative or positive for specific $k$. A more refined approach to
evaluating $\pi_{k}(x;m,n)-\pi(x^{k+1};m,n)$ when $x$ is sufficiently large
necessitates the utilization of the Riemann $\zeta$-function and the Riemann
Hypothesis to analyze the bias between $\pi(x;m,n)$ and
$\operatorname{li}(x)$, which allows for a more precise determination of the
error’s behavior in such cases.

##  2\. Proofs of Theorems

One useful technique used to prove Theorem 1.1 and Theorem 1.2 is Riemann-
Stieltjes Integration. The detailed procedure is included in Appendix 3.2.

###  2.1. Proof of Theorem 1.1

We divide the proof in three parts.

####  2.1.1. Start of Proof

###### Proof.

By the prime number theorem for arithmetic progressions [L, Theorem 7-25],

$\pi(x;m,n)\quad=\quad\frac{1}{\varphi(m)}\int_{2}^{x}\frac{\,dt}{\log t}\ +\
O\left(x\exp\left(-\frac{1}{2}\alpha\sqrt{\log x}\right)\right)\ (x\geq 2)$
(2.1)

where $\alpha$ is a positive constant. In the rest of the proof, denote

$E(x)\ =\ O\left(x\exp\left(-\frac{1}{2}\alpha\sqrt{\log x}\right)\right)$
(2.2)

as the error term in Equation (2.1).

By applying Equation (3.1) on Riemann-Stieltjes integration, for $x\geq 2$ we
have

$\pi_{k}(x;m,n)\quad=\sum_{\begin{subarray}{c}p\leq x\\\ p\ \equiv\ n\
(\mathrm{mod}\
m)\end{subarray}}p^{k}\quad=\quad\int_{2}^{x}t^{k}\,d\pi(t;m,n).$ (2.3)

Taking differentials yields

$\displaystyle d\pi(x;m,n)\ $ $\displaystyle=\
d\left(\frac{1}{\varphi(m)}\int_{2}^{x}\frac{\,dt}{\log t}\right)+dE(x)$
$\displaystyle=\ \frac{1}{\varphi(m)}\frac{\,dt}{\log t}+dE(x).$

Substituting $d\pi(x;m,n)$ into (2.3), we obtain

$\displaystyle\pi_{k}(x;m,n)\ $ $\displaystyle=\
\int_{2}^{x}t^{k}\left(\frac{1}{\varphi(m)}\frac{\,dt}{\log t}+dE(t)\right)$
$\displaystyle=\ \frac{1}{\varphi(m)}\int_{2}^{x}\frac{t^{k}}{\log
t}\,dt+\int_{2}^{x}t^{k}\,dE(t).$

The change of variable $t\mapsto t^{1/(k+1)}$ yields

$\displaystyle\frac{1}{\varphi(m)}\int_{2}^{x}\frac{t^{k}}{\log t}\,dt\ $
$\displaystyle=\
\frac{1}{\varphi(m)}\int_{2^{k+1}}^{x^{k+1}}\frac{t^{k/(k+1)}}{\log\left(t^{1/(k+1)}\right)}\frac{t^{-k/(k+1)}}{k+1}\,dt$
$\displaystyle=\ \frac{1}{\varphi(m)}\int_{2^{k+1}}^{x^{k+1}}\frac{\,dt}{\log
t}$ $\displaystyle=\ \pi(x^{k+1};m,n)-E(x^{k+1})-C$

where $C=\operatorname{li}(2^{k+1})/\varphi(m)$.

Thus,

$\frac{1}{\varphi(m)}\int_{2}^{x}\frac{t^{k}}{\log t}\,dt\ =\
\pi(x^{k+1};m,n)-E(x^{k+1})-C.$

Also, integration by parts yields

$\displaystyle\int_{2}^{x}t^{k}\,dE(t)\ $ $\displaystyle=\
t^{k}E(t)\Big{|}_{2}^{x}-k\int_{2}^{x}t^{k-1}E(t)\,dt$ $\displaystyle=\
x^{k}E(x)-k\int_{2}^{x}t^{k-1}E(t)\,dt.$

Finally, we obtain

$\displaystyle\pi_{k}(x;m,n)\ =\ $
$\displaystyle\pi(x^{k+1};m,n)-E(x^{k+1})-C+x^{k}E(x)-k\int_{2}^{x}t^{k-1}E(t)\,dt.$
(2.4)

####  2.1.2. Error Term

We now simplify (2.4).

First, according to (2.2),

$E(x)\ =\ O\left(x\exp\left(-\frac{1}{2}\alpha\sqrt{\log x}\right)\right).$

Thus,

$E(x^{k+1})\ =\ O\left(x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{(k+1)\log
x}\right)\right).$

Also, we have

$\displaystyle x^{k}E(x)\ $ $\displaystyle=\ x^{k}\
O\left(x\exp\left(-\frac{1}{2}\alpha\sqrt{\log x}\right)\right)$
$\displaystyle=\ O\left(x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{\log
x}\right)\right).$

The next step is to bound

$k\int_{2}^{x}t^{k-1}E(t)\,dt$ (2.5)

by using [GW, Lemma 1]. By applying (2.2), we can bound (2.5) as

$\displaystyle k\int_{2}^{x}t^{k-1}E(t)\,dt\ $ $\displaystyle\leq\
D\int_{2}^{x}t^{k-1}\left(t\exp\left(-\frac{1}{2}\alpha\sqrt{\log
t}\right)\right)\,dt$ $\displaystyle=\
D\int_{2}^{x}t^{k}\exp\left(-\frac{1}{2}\alpha\sqrt{\log t}\right)\,dt$ (2.6)

where $D$ is a positive constant.

The integrand of (2.6) is of the form $\displaystyle t^{k}\exp(-f(t))$ where
$\displaystyle f(t)=\frac{1}{2}\alpha\sqrt{\log t}$.
To apply [GW, Lemma 1], we verify that

  1. i.

$\displaystyle\lim_{t\to\infty}[t^{k+1}\exp(-f(t))]\ =\ \infty$

  2. ii.

and $\displaystyle\lim_{t\to\infty}[tf^{\prime}(t)]\ =\ 0$

for $k>-1$ in separate computations included in Appendix 3.3. Thus, applying
[GW, Lemma 1], we obtain

$k\int_{2}^{x}t^{k-1}E(t)\,dt\ =\
O\left(x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{\log x}\right)\right).$

Finally, we reduce

$O\left(x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{\log x}\right)\right)-C\ =\
O\left(x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{\log x}\right)\right).$

####  2.1.3. Conclusion

Based on previous sections, we simplify (2.4) as

$\displaystyle\pi_{k}$ $\displaystyle(x;m,n)-\pi(x^{k+1};m,n)$
$\displaystyle=\begin{cases}O\left(x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{\log
x}\right)\right)&\text{ if }k>0,\\\
O\left(x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{(k+1)\log
x}\right)\right)&\text{ if }-1<k<0.\end{cases}$

Therefore, we conclude that

$\pi_{k}(x;m,n)\ \sim\ \pi(x^{k+1};m,n).$

∎

###  2.2. Proof of Theorem 1.2

Before proving Theorem 1.2, we require additional results.

####  2.2.1. Abel’s Summation Formula and an Additional Lemma

_Abel’s summation formula_ is an important tool in analytic number theory
derived from integrating Riemann-Stieltjes by parts. The statement of the
formula and its hypotheses are given below.

First, recall that a function whose domain is the positive integers is called
an arithmetic function.

######  Theorem 2.1 (Abel’s Summation Formula).

Let $a$ be an arithmetic function. Define

$A(t)\ :=\ \sum\limits_{\begin{subarray}{c}1\leq n\leq t\end{subarray}}a(n)$

for $t\in\mathbb{R}$. Fix $x,y\in\mathbb{R}$ such that $x<y$ and let $f$ be a
continuously differentiable function on $[x,y]$. Then

$\sum\limits_{\begin{subarray}{c}x<n\leq y\end{subarray}}a(n)f(n)\ =\
A(y)f(y)-A(x)f(x)-\int_{x}^{y}A(t)f^{\prime}(t)\,dt.$ (2.7)

The following lemma can be found in Davenport’s Multiplicative Number Theory
textbook [D]. We provide a proof for that.

######  Lemma 2.1.

Fix positive integers $m,n\in\mathbb{Z}_{>0}$ such that $\gcd(m,n)=1$. Then
for $x\geq 2$,

$\sum\limits_{\begin{subarray}{c}p\leq x\\\ p\equiv
n\pmod{m}\end{subarray}}\frac{1}{p}\ =\
\frac{\log\log(x)}{\varphi(m)}+B+O\left(\frac{1}{\log x}\right),$

where $B$ is a constant.

###### Proof.

Let

$f(p)\ =\ \frac{1}{\log p}.$

Note that $f$ is defined on primes and is continuous on $[2,x]$. Let

$a(p)\ =\ \begin{cases}\frac{\log p}{p}&\text{ if}\quad p\ \equiv\
n\pmod{m}\quad\text{is prime},\\\ 0&\text{otherwise.}\end{cases}$

If $x\geq 2$,

$f^{\prime}(x)\ =\ -\frac{1}{x\log^{2}x}.$

Define

$A(x)\ :=\ \sum\limits_{\begin{subarray}{c}p\leq x\\\ p\equiv
n\pmod{m}\end{subarray}}a(p)\ =\ \sum\limits_{\begin{subarray}{c}p\leq x\\\
p\equiv n\pmod{m}\end{subarray}}\frac{\log p}{p}.$

According to [Ap, p.148, Theorem 7.3]

$\sum\limits_{\begin{subarray}{c}p\leq x\\\ p\equiv
n\pmod{m}\end{subarray}}\frac{\log p}{p}\ =\ \frac{\log x}{\varphi(m)}+R(x),\
\text{where}\ R(x)\ =\ O(1).$

Applying Abel’s Summation Formula (Theorem 2.1), we have

$\displaystyle\sum\limits_{\begin{subarray}{c}p\leq x\\\ p\equiv
n\pmod{m}\end{subarray}}\frac{1}{p}\ $ $\displaystyle=\
\sum\limits_{\begin{subarray}{c}1<p\leq x\\\ p\equiv
n\pmod{m}\end{subarray}}a(p)f(p)$ $\displaystyle=\ \frac{1}{\log
x}\left(\frac{\log
x}{\varphi(m)}+O(1)\right)+\int_{2}^{x}\frac{1}{t\log^{2}t}\left(\frac{\log
t}{\varphi(m)}+R(t)\right)\,dt$ $\displaystyle=\
\frac{1}{\varphi(m)}+O\left(\frac{1}{\log
x}\right)+\frac{1}{\varphi(m)}\int_{2}^{x}\frac{dt}{t\log
t}+\int_{2}^{x}\frac{R(t)}{t\log^{2}t}\,dt.$ (2.8)

Rewrite

$\int_{2}^{x}\frac{R(t)}{t\log^{2}t}dt\ =\
\int_{2}^{\infty}\frac{R(t)}{t\log^{2}t}\,dt-\int_{x}^{\infty}\frac{R(t)}{t\log^{2}t}\,dt.$

Notice that $\int_{2}^{\infty}\frac{R(t)}{t\log^{2}t}\,dt$ exists because
$R(t)=O(1).$

Further, since $R(t)=O(1)$, we have

$\int_{x}^{\infty}\frac{R(t)}{t\log^{2}t}\,dt\ =\
O\left(\int_{x}^{\infty}\frac{1}{t\log^{2}t}\,dt\right)\ =\
O\left(\frac{1}{\log x}\right).$

Continuing from (2.8):

$\displaystyle\sum\limits_{\begin{subarray}{c}p\leq x\\\ p\equiv
n\pmod{m}\end{subarray}}\frac{1}{p}\ $ $\displaystyle=\
\frac{1}{\varphi(m)}+O\left(\frac{1}{\log x}\right)+\frac{\log\log
x}{\varphi(m)}-\frac{\log\log 2}{\varphi(m)}$
$\displaystyle+\int_{2}^{\infty}\frac{R(t)}{t\log^{2}t}\,dt-\int_{x}^{\infty}\frac{R(t)}{t\log^{2}t}\,dt$
$\displaystyle=\ \frac{\log\log x}{\varphi(m)}+B+O\left(\frac{1}{\log
x}\right),$

where

$B\ =\ \frac{1}{\varphi(m)}-\frac{\log\log
2}{\varphi(m)}+\int_{2}^{\infty}\frac{R(t)}{t\log^{2}t}\,dt.$

Therefore, we conclude that

$\sum\limits_{\begin{subarray}{c}p\leq x\\\ p\equiv
n\pmod{m}\end{subarray}}\frac{1}{p}\ =\ \frac{\log\log
x}{\varphi(m)}+B+O\left(\frac{1}{\log x}\right).$

∎

####  2.2.2. Proof of Theorem 1.2 by Lemma 2.1

Recall the statement of Theorem 1.2.

######  Theorem.

1.2 Fix a real number $k>-1$ and positive integers $m,n\in\mathbb{Z}_{>0}$
such that $\gcd(m,n)=1$. Then

$\int_{1}^{\infty}\frac{\pi_{k}(t;m,n)-\pi(t^{k+1};m,n)}{t^{k+2}}\,dt\ =\
-\frac{\log(k+1)}{(k+1)\varphi(m)}.$

Further,

$\displaystyle-\frac{\log(k+1)}{(k+1)\varphi(m)}\ $ $\displaystyle<\
0\qquad(k>0)$ $\displaystyle-\frac{\log(k+1)}{(k+1)\varphi(m)}\ $
$\displaystyle>\ 0\qquad(-1<k<0).$

###### Proof.

By Lemma 2.1, if $x\geq 2$, we have

$\displaystyle\sum\limits_{\begin{subarray}{c}p\leq x\\\ p\equiv
n\pmod{m}\end{subarray}}\frac{1}{p}$ $\displaystyle\ =\
\frac{\log\log(x)}{\varphi(m)}+B+O\left(\frac{1}{\log x}\right)$
$\displaystyle=\ \frac{1}{\varphi(m)}\log\log x+B+o(1),$

where $B$ is a constant.

Let $u=t^{k+1}$. Then

$\displaystyle\int_{1}^{x}\frac{\pi(t^{k+1};m,n)}{t^{k+2}}\,dt$
$\displaystyle\ =\
\frac{1}{k+1}\int_{1}^{x^{k+1}}\frac{\pi(u;m,n)}{u^{2}}\,du$ $\displaystyle=\
\frac{-1}{k+1}\frac{\pi(x^{k+1};m,n)}{x^{k+1}}+\frac{1}{k+1}\int_{1}^{x^{k+1}}\frac{1}{u}\,d\pi(u;m,n)$
$\displaystyle=\
\frac{-1}{k+1}\frac{\pi(x^{k+1};m,n)}{x^{k+1}}+\frac{1}{k+1}\sum\limits_{\begin{subarray}{c}p\leq
x^{k+1}\\\ p\equiv n\pmod{m}\end{subarray}}\frac{1}{p}$ $\displaystyle=\
\frac{\log\log(x^{k+1})}{(k+1)\varphi(m)}+\frac{B}{k+1}+o(1).$

Similarly,

$\displaystyle\int_{1}^{x}\frac{\pi_{k}(t,m,n)}{t^{k+2}}\,dt$ $\displaystyle\
=\
\frac{-1}{k+1}\frac{\pi_{k}(x,m,n)}{x^{k+1}}+\frac{1}{k+1}\int_{1}^{x}\frac{1}{t^{k+1}}\,d\pi_{k}(t,m,n)$
$\displaystyle=\
\frac{-1}{k+1}\frac{\pi_{k}(x,m,n)}{x^{k+1}}+\frac{1}{k+1}\sum\limits_{\begin{subarray}{c}p\leq
x\\\ p\equiv n\pmod{m}\end{subarray}}\frac{1}{p^{k+1}}p^{k}$ $\displaystyle=\
\frac{-1}{k+1}\frac{\pi_{k}(x,m,n)}{x^{k+1}}+\frac{1}{k+1}\sum\limits_{\begin{subarray}{c}p\leq
x\\\ p\equiv n\pmod{m}\end{subarray}}\frac{1}{p}$ $\displaystyle=\
\frac{\log\log(x^{k+1})}{(k+1)\varphi(m)}+\frac{B}{k+1}+o(1).$

Thus,

$\int_{1}^{x}\frac{\pi_{k}(t,m,n)-\pi(t^{k+1},m,n)}{t^{k+2}}\,dt\ =\
-\frac{\log(k+1)}{(k+1)\varphi(m)}+o(1),$

which implies the theorem. ∎

##  3\. Appendices

###  3.1. Appendix: Some Examples of Main Result

In this section, we present tables containing illustrative examples that
showcase the accuracy of our main results. The main goal is to view how
closely $\pi_{k}(x^{1/(k+1)};m,n)$ approximates $\pi(x;m,n)$.

Utilizing High Performance Computing111Appendix 3.1 is based upon work
supported by the Great Lakes Slurm cluster at the University of Michigan, Ann
Arbor, MI, where the author (Zhangze Li) is currently enrolled as an
undergraduate student, and the National Science Foundation under Grant No.
DMS-1929284 while the author (Stephanie Reyes) was in residence at the
Institute for Computational and Experimental Research in Mathematics in
Providence, RI, during the Summer@ICERM program., we examine the approximation
for primes of the form $p\equiv 1\pmod{4}$, $p\equiv 3\pmod{4}$, $p\equiv
1\pmod{5}$, and $p\equiv 3\pmod{5}$. With above modular cases, we select $9$
positive integers between $10^{4}$ and $10^{8}$, and calculate

$\text{Error}\ (x,k;m,n)\ :=\
\frac{\pi(x;m,n)-\pi_{k}(x^{1/(k+1)};m,n)}{\pi(x;m,n)}$

for $k=1,\ 1/2,\ -1/10,\ -1/12$.

####  3.1.1. Error for $k=1$

We observe that of the 36 Errors measured in Tables 1-4, $72.22\%$ are
positive.

$x$ | $\pi(x;4,1)$ | $\pi_{1}(x^{1/2};4,1)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 609 | 515 | $15.43514\%$
$5\times 10^{4}$ | 2549 | 2025 | $20.55708\%$
$1\times 10^{5}$ | 4783 | 4418 | $7.63119\%$
$5\times 10^{5}$ | 20731 | 19668 | $5.12759\%$
$1\times 10^{6}$ | 39175 | 36628 | $6.50160\%$
$5\times 10^{6}$ | 174193 | 165373 | $5.06335\%$
$1\times 10^{7}$ | 332180 | 323048 | $2.74911\%$
$5\times 10^{7}$ | 1500452 | 1475230 | $1.68096\%$
$1\times 10^{8}$ | 2880504 | 2863281 | $0.59792\%$
Table 1. $\pi(x;4,1)$ and $\pi_{1}(x^{1/2};4,1)$. $x$ | $\pi(x;4,3)$ | $\pi_{1}(x^{1/2};4,3)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 619 | 543 | $12.27787\%$
$5\times 10^{4}$ | 2583 | 2411 | $6.65892\%$
$1\times 10^{5}$ | 4808 | 4786 | $0.45757\%$
$5\times 10^{5}$ | 20806 | 20643 | $0.78343\%$
$1\times 10^{6}$ | 39322 | 39497 | $-0.44504\%$
$5\times 10^{6}$ | 174319 | 170667 | $2.09501\%$
$1\times 10^{7}$ | 332398 | 319819 | $3.78432\%$
$5\times 10^{7}$ | 1500681 | 1516507 | $-1.05459\%$
$1\times 10^{8}$ | 2880950 | 2873113 | $0.27203\%$
Table 2. $\pi(x;4,3)$ and $\pi_{1}(x^{1/2};4,3)$. $x$ | $\pi(x;5,1)$ | $\pi_{1}(x^{1/2};5,1)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 306 | 215 | $29.73856\%$
$5\times 10^{4}$ | 1274 | 1181 | $7.29984\%$
$1\times 10^{5}$ | 2387 | 2536 | $-6.24214\%$
$5\times 10^{5}$ | 10386 | 10631 | $-2.35894\%$
$1\times 10^{6}$ | 19617 | 18470 | $5.84697\%$
$5\times 10^{6}$ | 87062 | 81830 | $6.00951\%$
$1\times 10^{7}$ | 166104 | 156148 | $5.99384\%$
$5\times 10^{7}$ | 750340 | 763457 | $-1.74814\%$
$1\times 10^{8}$ | 1440298 | 1448386 | $-0.56155\%$
Table 3. $\pi(x;5,1)$ and $\pi_{1}(x^{1/2};5,1)$. $x$ | $\pi(x;5,3)$ | $\pi_{1}(x^{1/2};5,3)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 310 | 291 | $6.12903\%$
$5\times 10^{4}$ | 1290 | 1036 | $19.68992\%$
$1\times 10^{5}$ | 2402 | 2644 | $-10.07494\%$
$5\times 10^{5}$ | 10382 | 10539 | $-1.51223\%$
$1\times 10^{6}$ | 19665 | 18146 | $7.72438\%$
$5\times 10^{6}$ | 87216 | 90461 | $-3.72065\%$
$1\times 10^{7}$ | 166230 | 156456 | $5.87981\%$
$5\times 10^{7}$ | 750395 | 753820 | $-0.45643\%$
$1\times 10^{8}$ | 1440474 | 1436510 | $0.27519\%$
Table 4. $\pi(x;5,3)$ and $\pi_{1}(x^{1/2};5,3)$.

####  3.1.2. Error for $k=1/2$

We observe that of the 36 Errors measured in Tables 5-8, $72.22\%$ are
positive.

$x$ | $\pi(x;4,1)$ | $\pi_{1/2}(x^{2/3};4,1)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 609 | 617.62512 | $-1.41628\%$
$5\times 10^{4}$ | 2549 | 2477.64505 | $2.79933\%$
$1\times 10^{5}$ | 4783 | 4659.83812 | $2.57499\%$
$5\times 10^{5}$ | 20731 | 20125.89212 | $2.91886\%$
$1\times 10^{6}$ | 39175 | 38904.00140 | $0.69176\%$
$5\times 10^{6}$ | 174193 | 173246.23939 | $0.54351\%$
$1\times 10^{7}$ | 332180 | 329252.45078 | $0.88131\%$
$5\times 10^{7}$ | 1500452 | 1492885.30185 | $0.50429\%$
$1\times 10^{8}$ | 2880504 | 2873027.62482 | $0.25955\%$
Table 5. $\pi(x;4,1)$ and $\pi_{1/2}(x^{2/3};4,1)$. $x$ | $\pi(x;4,3)$ | $\pi_{1/2}(x^{2/3};4,3)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 619 | 591.60159 | $4.42624\%$
$5\times 10^{4}$ | 2583 | 2502.18366 | $3.12878\%$
$1\times 10^{5}$ | 4808 | 4833.35209 | $-0.52729\%$
$5\times 10^{5}$ | 20806 | 20951.89316 | $-0.70121\%$
$1\times 10^{6}$ | 39322 | 39178.87051 | $0.36399\%$
$5\times 10^{6}$ | 174319 | 173924.73741 | $0.22617\%$
$1\times 10^{7}$ | 332398 | 331806.98445 | $0.17780\%$
$5\times 10^{7}$ | 1500681 | 1502046.79913 | $-0.09101\%$
$1\times 10^{8}$ | 2880950 | 2879155.53993 | $0.06229\%$
Table 6. $\pi(x;4,3)$ and $\pi_{1/2}(x^{2/3};4,3)$. $x$ | $\pi(x;5,1)$ | $\pi_{1/2}(x^{2/3};5,1)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 306 | 290.30286 | $5.12978\%$
$5\times 10^{4}$ | 1274 | 1243.85408 | $2.36624\%$
$1\times 10^{5}$ | 2387 | 2288.69057 | $4.11853\%$
$5\times 10^{5}$ | 10386 | 10309.63049 | $0.73531\%$
$1\times 10^{6}$ | 19617 | 19616.30635 | $0.00354\%$
$5\times 10^{6}$ | 87062 | 87036.61969 | $0.02915\%$
$1\times 10^{7}$ | 166104 | 164310.69864 | $1.07963\%$
$5\times 10^{7}$ | 750340 | 752249.09877 | $-0.25443\%$
$1\times 10^{8}$ | 1440298 | 1430300.15946 | $0.69415\%$
Table 7. $\pi(x;5,1)$ and $\pi_{1/2}(x^{2/3};5,1)$. $x$ | $\pi(x;5,3)$ | $\pi_{1/2}(x^{2/3};5,3)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 310 | 321.30898 | $-3.64806\%$
$5\times 10^{4}$ | 1290 | 1244.44539 | $3.53136\%$
$1\times 10^{5}$ | 2402 | 2501.69252 | $-4.15040\%$
$5\times 10^{5}$ | 10382 | 10504.71338 | $-1.18198\%$
$1\times 10^{6}$ | 19665 | 19604.34768 | $0.30843\%$
$5\times 10^{6}$ | 87216 | 86272.37280 | $1.08194\%$
$1\times 10^{7}$ | 166230 | 167998.78804 | $-1.06406\%$
$5\times 10^{7}$ | 750395 | 748916.40906 | $0.19704\%$
$1\times 10^{8}$ | 1440474 | 1444351.68992 | $-0.26920\%$
Table 8. $\pi(x;5,3)$ and $\pi_{1/2}(x^{2/3};5,3)$.

####  3.1.3. Error for $k=-1/10$

We observe that of the 36 Errors measured in Tables 9-12, $72.22\%$ are
negative.

$x$ | $\pi(x;4,1)$ | $\pi_{-1/10}(x^{10/9};4,1)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 609 | 613.50169 | $-0.73919\%$
$5\times 10^{4}$ | 2549 | 2562.89963 | $-0.54530\%$
$1\times 10^{5}$ | 4783 | 4788.03485 | $-0.10527\%$
$5\times 10^{5}$ | 20731 | 20771.18437 | $-0.19384\%$
$1\times 10^{6}$ | 39175 | 39266.51644 | $-0.23361\%$
$5\times 10^{6}$ | 174193 | 174232.64634 | $-0.02276\%$
$1\times 10^{7}$ | 332180 | 332314.25320 | $-0.04042\%$
$5\times 10^{7}$ | 1500452 | 1500545.39963 | $-0.00622\%$
$1\times 10^{8}$ | 2880504 | 2880813.47274 | $-0.01074\%$
Table 9. $\pi(x;4,1)$ and $\pi_{-1/10}(x^{10/9};4,1)$. $x$ | $\pi(x;4,3)$ | $\pi_{-1/10}(x^{10/9};4,3)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 619 | 618.68563 | $0.05079\%$
$5\times 10^{4}$ | 2583 | 2579.32406 | $0.14231\%$
$1\times 10^{5}$ | 4808 | 4821.56790 | $-0.28219\%$
$5\times 10^{5}$ | 20806 | 20796.90807 | $0.04370\%$
$1\times 10^{6}$ | 39322 | 39305.82276 | $0.04114\%$
$5\times 10^{6}$ | 174319 | 174323.13838 | $-0.00237\%$
$1\times 10^{7}$ | 332398 | 332477.68215 | $-0.02397\%$
$5\times 10^{7}$ | 1500681 | 1500779.09018 | $-0.00654\%$
$1\times 10^{8}$ | 2880950 | 2881063.76609 | $-0.00395\%$
Table 10. $\pi(x;4,3)$ and $\pi_{-1/10}(x^{10/9};4,3)$. $x$ | $\pi(x;5,1)$ | $\pi_{-1/10}(x^{10/9};5,1)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 306 | 306.84917 | $-0.27750\%$
$5\times 10^{4}$ | 1274 | 1284.72538 | $-0.84187\%$
$1\times 10^{5}$ | 2387 | 2397.35493 | $-0.43380\%$
$5\times 10^{5}$ | 10386 | 10381.05165 | $-0.04764\%$
$1\times 10^{6}$ | 19617 | 19624.30360 | $-0.03723\%$
$5\times 10^{6}$ | 87062 | 87119.41013 | $-0.06594\%$
$1\times 10^{7}$ | 166104 | 166161.85950 | $-0.03483\%$
$5\times 10^{7}$ | 750340 | 750274.35740 | $0.00875\%$
$1\times 10^{8}$ | 1440298 | 1440380.00374 | $-0.00569\%$
Table 11. $\pi(x;5,1)$ and $\pi_{-1/10}(x^{10/9};5,1)$. $x$ | $\pi(x;5,3)$ | $\pi_{-1/10}(x^{10/9};5,3)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 310 | 309.65259 | $0.11207\%$
$5\times 10^{4}$ | 1290 | 1288.73787 | $0.09784\%$
$1\times 10^{5}$ | 2402 | 2406.10885 | $-0.17106\%$
$5\times 10^{5}$ | 10382 | 10404.21103 | $-0.21394\%$
$1\times 10^{6}$ | 19665 | 19658.64066 | $0.03234\%$
$5\times 10^{6}$ | 87216 | 87152.03182 | $0.07334\%$
$1\times 10^{7}$ | 166230 | 166229.30179 | $0.00042\%$
$5\times 10^{7}$ | 750395 | 750428.14712 | $-0.00442\%$
$1\times 10^{8}$ | 1440474 | 1440534.90175 | $-0.00423\%$
Table 12. $\pi(x;5,3)$ and $\pi_{-1/10}(x^{10/9};5,3)$.

####  3.1.4. Error for $k=-1/12$

We observe that of the 36 Errors measured in Tables 13-16, $77.78\%$ are
negative.

$x$ | $\pi(x;4,1)$ | $\pi_{-1/12}(x^{12/11};4,1)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 609 | 611.17719 | $-0.35750\%$
$5\times 10^{4}$ | 2549 | 2558.04851 | $-0.35498\%$
$1\times 10^{5}$ | 4783 | 4787.40169 | $-0.09203\%$
$5\times 10^{5}$ | 20731 | 20762.97056 | $-0.15422\%$
$1\times 10^{6}$ | 39175 | 39253.59228 | $-0.20062\%$
$5\times 10^{6}$ | 174193 | 174211.69780 | $-0.01073\%$
$1\times 10^{7}$ | 332180 | 332319.22077 | $-0.04191\%$
$5\times 10^{7}$ | 1500452 | 1500497.92446 | $-0.00306\%$
$1\times 10^{8}$ | 2880504 | 2880766.12283 | $-0.00910\%$
Table 13. $\pi(x;4,1)$ and $\pi_{-1/12}(x^{12/11};4,1)$. $x$ | $\pi(x;4,3)$ | $\pi_{-1/12}(x^{12/11};4,3)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 619 | 622.36367 | $-0.54340\%$
$5\times 10^{4}$ | 2583 | 2584.93696 | $-0.07499\%$
$1\times 10^{5}$ | 4808 | 4816.96821 | $-0.18653\%$
$5\times 10^{5}$ | 20806 | 20788.86077 | $0.08238\%$
$1\times 10^{6}$ | 39322 | 39306.76238 | $0.03875\%$
$5\times 10^{6}$ | 174319 | 174325.21822 | $-0.00357\%$
$1\times 10^{7}$ | 332398 | 332443.67182 | $-0.01374\%$
$5\times 10^{7}$ | 1500681 | 1500833.67573 | $-0.01017\%$
$1\times 10^{8}$ | 2880950 | 2881059.64847 | $-0.00381\%$
Table 14. $\pi(x;4,3)$ and $\pi_{-1/12}(x^{12/11};4,3)$. $x$ | $\pi(x;5,1)$ | $\pi_{-1/12}(x^{12/11};5,1)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 306 | 308.32261 | $-0.75902\%$
$5\times 10^{4}$ | 1274 | 1286.24176 | $-0.96089\%$
$1\times 10^{5}$ | 2387 | 2392.89814 | $-0.24709\%$
$5\times 10^{5}$ | 10386 | 10367.25069 | $0.18052\%$
$1\times 10^{6}$ | 19617 | 19631.27471 | $-0.07277\%$
$5\times 10^{6}$ | 87062 | 87117.75683 | $-0.06404\%$
$1\times 10^{7}$ | 166104 | 166145.81965 | $-0.02518\%$
$5\times 10^{7}$ | 750340 | 750274.07185 | $0.00879\%$
$1\times 10^{8}$ | 1440298 | 1440333.42281 | $-0.00246\%$
Table 15. $\pi(x;5,1)$ and $\pi_{-1/12}(x^{12/11};5,1)$. $x$ | $\pi(x;5,3)$ | $\pi_{-1/12}(x^{12/11};5,3)$ | Error $\%$
---|---|---|---
$1\times 10^{4}$ | 310 | 310.43590 | $-0.14061\%$
$5\times 10^{4}$ | 1290 | 1290.55374 | $-0.04293\%$
$1\times 10^{5}$ | 2402 | 2406.74513 | $-0.19755\%$
$5\times 10^{5}$ | 10382 | 10393.81954 | $-0.11385\%$
$1\times 10^{6}$ | 19665 | 19662.29214 | $0.01377\%$
$5\times 10^{6}$ | 87216 | 87159.52414 | $0.06475\%$
$1\times 10^{7}$ | 166230 | 166209.29782 | $0.01245\%$
$5\times 10^{7}$ | 750395 | 750355.12871 | $0.00531\%$
$1\times 10^{8}$ | 1440474 | 1440579.59405 | $-0.00733\%$
Table 16. $\pi(x;5,3)$ and $\pi_{-1/12}(x^{12/11};5,3)$.

####  3.1.5. Conclusion

Note that when $k\ >\ 0$, Error exhibits a positive trend, indicating that
$\pi_{(}x^{k+1};m,n)$ tends to be larger $\pi_{k}(x;m,n)$. The opposite trend
holds when $k\ <\ 0$. The explanation for this phenomenon is provided by
Theorem 1.2 and Remark 1.

###  3.2. Appendix: Riemann–Stieltjes Integration

####  3.2.1. Riemann-Stieltjes Integral

Let $f$ be a real-valued function of a real variable on $[a,b]$ where
$a,b\in\mathbb{R}$, and let $g$ also be a real-to-real function. Let
$P=\\{a=x_{0}<x_{1}<x_{2}<\dots<x_{n}=b\\}$ be a partition of $[a,b]$.
Consider

$S(f,g,P)\ :=\ \sum_{i=1}^{n}f(x_{i}^{*})[g(x_{i})-g(x_{i-1})]$

where $x_{i}^{*}\in[x_{i-1},x_{i}].$

If $S(f,g,P)$ converges to a constant $L$ when $\max\limits_{1\leq i\leq
n}\\{x_{i}-x_{i-1}\\}$ approaches $0$, then we define Riemann-Stieltjes
Integral as following:

$\int_{a}^{b}f\,dg\ :=\ L.$

The Riemann-Steltjes integral exists if $g$ is of bounded variation and right-
semicontinuous.

####  3.2.2. Applying Riemann-Stieltjes Integration

Consider

$\pi(x;m,n)-\pi(x-1;m,n)\ =\ \begin{cases}1,&\text{if $x$ is prime and $x\
\equiv\ n\ (\mathrm{mod}\ m)$},\\\ \\\
0,&\begin{subarray}{c}\displaystyle\text{if $x$ is composite or}\\\
\displaystyle\text{$x$ is prime where $x\ \not\equiv\ n\ (\mathrm{mod}\
m)$.}\end{subarray}\end{cases}$

Let $f$ be a real-valued function of a real variable, then

$\sum_{\begin{subarray}{c}p\leq x\\\ p\ \equiv\ n\ (\mathrm{mod}\
m)\end{subarray}}f(p)\qquad=\qquad\sum_{i=2}^{x}f(i)[\pi(x;m,n)-\pi(x-1;m,n)].$

By applying the Riemann-Stieltjes Integral, we have

$\displaystyle\sum_{\begin{subarray}{c}p\leq x\\\ p\ \equiv\ n\ (\mathrm{mod}\
m)\end{subarray}}f(p)\qquad$
$\displaystyle=\qquad\sum_{i=2}^{x}f(i)[\pi(x;m,n)-\pi(x-1;m,n)]$
$\displaystyle=\qquad\int_{2}^{x}f(x)\,d\pi(x;m,n).$ (3.1)

This technique is crucial to evaluate the Riemann-Stieltjes Integral in proofs
of Theorem 1.1 and Theorem 1.2.

###  3.3. Appendix: Limit Computation

From Gerard-Washington [GW, Lemma 1], we have the following result.

Let $k>-1$ and let $f$ be a differentiable function on $[2,\infty)$ such that

  1. i.

$\displaystyle x^{k+1}\exp(-f(x))\to\infty$ and

  2. ii.

$\displaystyle xf^{\prime}(x)\to 0$

as $x\to\infty$. Then $\displaystyle\int_{2}^{x}t^{k}\exp(-f(t))\,dt\ =\
O\left(x^{k+1}\exp(-f(x))\right)$.

We apply this result with $f(x)=\frac{1}{2}\alpha\sqrt{\log x}$, where
$\alpha$ is a positive constant.

We verified the hypotheses in our case by computing the relevant limits for
applying the lemma.

  1. i.

We will find

$L\ :=\ \lim_{x\to\infty}\left[x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{\log
x}\right)\right].$

Let

$x\ =\ \exp\left(\frac{4y^{2}}{\alpha^{2}}\right).$

When $y\to\infty$, $x\to\infty$. Thus,

$\displaystyle L\ $ $\displaystyle=\
\lim_{y\to\infty}\left[\left[\exp\left(\frac{4y^{2}}{\alpha^{2}}\right)\right]^{k+1}\exp\left(-\frac{1}{2}\alpha\cdot\frac{2y}{\alpha}\right)\right]$
$\displaystyle=\
\lim_{y\to\infty}\left[\exp\left((k+1)\cdot\frac{4y^{2}}{\alpha^{2}}\right)\exp\left(-y\right)\right]$
$\displaystyle=\ \lim_{y\to\infty}\left[\exp\left(\frac{4k+4}{\alpha^{2}}\cdot
y^{2}\right)\exp\left(-y\right)\right].$

Since $k\ >\ -1$, we have $k\ +\ 1\ >\ 0\Longrightarrow\ 4k\ +\ 4\ >\ 0.$
Hence,

$\frac{4k+4}{\alpha^{2}}\ >\ 0,$

which indicates that

$y\ \lll\frac{4k+4}{\alpha^{2}}\cdot y^{2}$

when $y$ approaches $\infty$.

Thus, $\exp\left(\frac{4k+4}{\alpha^{2}}\cdot y^{2}\right)$ grows much faster
than $\exp\left(-y\right)$ decays, so we conclude that

$L\ =\ \lim_{y\to\infty}\left[\exp\left(\frac{4k+4}{\alpha^{2}}\cdot
y^{2}\right)\exp\left(-y\right)\right]\ =\ \infty.$

Therefore,

$\lim_{x\to\infty}\left[x^{k+1}\exp\left(-\frac{1}{2}\alpha\sqrt{\log
x}\right)\right]\ =\ \infty.$

  2. ii.

Recall that $\displaystyle f(x)\ =\ \frac{1}{2}\alpha\sqrt{\log x}$. Thus

$\displaystyle f^{\prime}(x)\ $ $\displaystyle=\ \frac{\alpha}{4x\sqrt{\log
x}}$ $\displaystyle\Longrightarrow\qquad xf^{\prime}(x)\ $ $\displaystyle=\
\frac{\alpha}{4\sqrt{\log x}}$

Since $\displaystyle\lim_{x\to\infty}\sqrt{\log x}\ =\ \infty$, it follows
that

$\lim_{x\to\infty}[xf^{\prime}(x)]\ =\ 0.$

## References

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