# Hong-Ou-Mandel interference with a diode-pumped 1-GHz Ti:sapphire laser

Imogen Morland, Hanna Ostapenko, Feng Zhu, Derryck T. Reid and Jonathan Leach∗
Institute of Photonics and Quantum Sciences, Heriot-Watt University, David
Brewster Building, Edinburgh EH14 4AS, UK<EMAIL_ADDRESS>
###### Abstract

Correlated photon pairs generated through spontaneous parametric down-
conversion (SPDC) are a key resource in quantum optics. In many quantum optics
applications, such as satellite quantum key distribution (QKD), a compact,
high repetition rate pump laser is required. Here we demonstrate the use of a
compact, GHz-rate diode-pumped three-element Kerr-lens-modelocked Ti:sapphire
laser for the generation of correlated photon pairs at 790 nm. We verify the
presence of indistinguishable photons produced via SPDC using Hong-Ou-Mandel
(HOM) interferometry and observe a dip in coincidence counts with a visibility
of 81.8%.

##  I Introduction

Hong-Ou-Mandel (HOM) interference hong1987measurement  is a two-photon effect
that demonstrates the quantum nature of indistinguishable single photons. It
has applications in quantum optics shih1988new ; rarity1990two ;
santori2002indistinguishable , quantum communication ekert1991quantum ;
yin2017satellite  and quantum computing kok2007linear . HOM interference also
plays an important role in quantum metrology, where quantum phenomena such as
entanglement are used boto2000quantum ; giovannetti2011advances . For example,
the generation of photon pairs via spontaneous parametric down-conversion
(SPDC) can be used as a source for N00N states dowling2008quantum . These
states are necessary to achieve the fundamental quantum limit for phase
sensitivity known as the Heisenberg limit bouchard2020two . HOM interference
also plays a key role in Bell-state measurements weinfurter1994experimental ,
which are used in entanglement swapping zhang2016engineering ;
zhang2017simultaneous  and quantum teleportation bouwmeester1997experimental .
Furthermore, quantum optical coherence tomography (QOCT) nasr2003demonstration
builds on optical coherence tomography (OCT) huang1991optical  by using HOM
interferometry to measure the depth profile of biological samples to
micrometre resolution abouraddy2002quantum ; nasr2004dispersion ;
lopez2012quantum . HOM interferometry has been used in the high precision
measurement of time delays dauler1999tests ; branning2000simultaneous  and
polarization harnchaiwat2020tracking  and when combined with statistical
estimation theory, nanometer precision has been achieved lyons2018attosecond .
More recently, HOM interferometry was used to image micrometre-scale depth
features ndagano2022quantum .

A photon pair source with high brightness, visibility and collection
efficiency combined with a high generation rate is required in many quantum
optics applications bouwmeester1997experimental ; liao2017satellite . High
brightness integrated sources have been demonstrated vergyris2017fully ;
fan2007bright ; sansoni2017two  and there are many examples where the
brightness, visibility and collection efficiency are close to optimal
zhong201812 ; meyer2018high . A domain-engineering technique for tailoring the
crystal nonlinearity to generate indistinguishable and spectrally pure photons
without filtering has also been demonstrated graffitti2018independent .

The development of GHz ultrafast lasers has opened the door for high
generation rate photon pair sources, and such GHz lasers have been used to
generate SPDC at telecommunication wavelengths zhang2008generation ;
jin2014efficient ; ngah2015ultra . Recently, the HOM interference between two
photons from two independent 10 GHz sources, spaced by distances of up to 100
km has been demonstrated with visibilities in excess of 90% d2020universal .
Further work has been done to improve the indistinguishability of photon pairs
produced via SPDC with a GHz laser using temporal filtering techniques
miyanishi2020robust . The effect of GHz repetition rate lasers on the spectral
purity has been modelled and confirmed experimentally by demonstrating high-
visibility HOM interference between two independent heralded single photons
(HSPs) generated by SPDC with $3.2$ GHz pump pulses tsujimoto2021ultra .

One of the issues with the generation of quantum states of light is that
typical ultrafast lasers are bulky and expensive. Here we demonstrate the use
of a compact, low cost GHz laser ostapenko2022three ; ostapenko2023design ,
with characteristics compatible for applications where size, weight and power
are important, such as satellite quantum key distribution (QKD)
liao2017satellite . We show that this compact GHz laser can be used to produce
indistinguishable photon pairs for quantum optics as demonstrated by the
observation of a high visibility HOM dip.

##  II Experimental Set-up

Figure 1: The experimental setup has two parts: the laser cavity and the
Hong–Ou–Mandel (HOM) experiment. The output beam in (a) is the input beam in
(b). (a) The laser cavity was pumped by a 1.1 W laser diode (Nichia NDG7D75),
which is collimated via three lenses ($L_{1}$, $L_{2}$ and $L_{3}$), and
focused onto the Ti:sapphire gain crystal using a plano-convex achromatic
doublet lens with a focal length of 45 mm ($L_{4}$). The oscillator includes a
concave mirror ($M_{1}$) with R = -50 mm and a plane end mirror ($M_{2}$). The
beams are separated using a dichroic mirror (D) and focused into a $\beta$-BBO
with a concave mirror $M_{3}$ with R = -75 mm and 395 nm pulses are produced.
A series of lenses and cylindrical lenses were used to collimate the beam and
reduce the beam radius. (b) The up-converted light was filtered using a
bandpass filter, $F_{1}$ (Thorlabs FBH405-10) and then focused into a second
Type I $\beta$-BBO and the signal and idler photons were imaged via lenses
($L_{6}$ and $L_{7}$) with focal lengths $f_{1}~{}=150$ mm and $f_{2}~{}=60$
mm. The signal and idler photons were filtered using a tilted 790 nm bandpass
filter $F_{2}$ (Thorlabs FBH790-10) and directed down different paths using a
D shaped mirror (DSM) and two pinholes were placed in each of the paths. The
down-converted photons are coupled into a SM FBS, each with lenses of focal
length 7.5 mm, via mirrors and a prism (P). Single photon counts were recorded
by each SPAD (Excelitas SPCM-NIR), and from this coincidences counts are
calculated as a function of the translation stage position, z.

Our generation and measurement system comprises three stages. In the first
stage, a laser diode was used to pump the three-element Kerr-lens-modelocked
Ti:sapphire laser and 790 nm, 105 fs pulses were produced at 1 GHz. Next,
these laser pulses were frequency doubled using a type I $\beta$-BBO crystal
to produce 395 nm light. Then, a second $\beta$-BBO crystal was used to
produce photon pairs via SPDC, which traveled through each arm of the HOM
interferometer and were coupled into a single-mode fiber beam splitter (SM
FBS). Coincidence counts were then recorded by single-photon avalanche diodes
(SPADs) as a function of the relative optical delay between the photon pairs.
The visibility of the dip in coincidences confirms the generation of
indistinguishable single photons, when it exceeds 50%.

The laser platform used for the experiment was a diode pumped three-element
Kerr-lens-modelocked Ti:sapphire cavity ostapenko2022three ;
ostapenko2023design  as shown in Fig. 1 (a). A single laser diode (Nichia
NDG7D75) operated at a maximum pump power of 1.1 W was used to pump the
cavity. The fast axis of the diode was collimated using $L_{1}$ and the slow
axis was expanded using two cylindrical lenses ($L_{2}$ and $L_{3}$), which
act as a 1:6 telescope.

The Ti:sapphire crystal has an absorption of 4.1 cm-1 at 532 nm, a figure of
merit ¿200, and a plane-Brewster geometry where the plane side of the mirror
is coated to be transmitting at 450-530 nm and 99% reflective at 770-830 nm.
The coating specifications allow the plane side of the crystal to act as both
the pump in-coupling mirror for the cavity and the output coupler of 1%. A
focusing achromatic doublet lens, $L_{4}$, with 45 mm focal length was used to
both focus the pump beam into the crystal and collimate the laser output, and
the two beams were separated by a dichroic mirror (D) outside the cavity.

The oscillator itself also includes a concave mirror with R = -50 mm and a
plane end mirror with a Gires-Tournois interferometer (GTI) coating of -550
fs2 used for dispersion optimisation of the cavity. The 790 nm output beam was
then separated from the pump using dichroic mirror D and focused into a 2 mm
thick $\beta$-BBO crystal with a focusing mirror $M_{3}$ with R = -75 mm and
second-harmonic generation (SHG) pulses were produced. A series of lenses and
cylindrical lenses collimated the beam and reduced it’s radius.

The HOM experiment is shown in Fig 1 (b). The up-converted pulses were
filtered using a bandpass filter angle-tuned to 395 nm with a full width at
half maximum (FWHM) of 10 nm (Thorlabs FBH405-10), $F_{1}$, and the beam was
focused into another Type I $\beta$-BBO via lens $L_{6}$ with focal length
$f_{1}~{}=150$ mm. In the alignment stage, the photon pairs generated via SPDC
were relayed using an electron multiplying charge-coupled device (EMCCD) and
the angle of the $\beta$-BBO crystal was adjusted to give the desired phase
matching condition. Two pinholes were then positioned in each arm of the HOM
interferometer such that the signal and idler photons were selected and an
infrared (IR) laser diode was used for back projection. This was done using
two alignment mirrors in each arm such that the light from the IR laser diode
propagated through the pinholes. The coincidence counts were coupled into
single-mode (SM) fibers using 7.5 mm focal length lenses and were optimized
with alignment mirrors and the three axis fiber alignment stages. A 650 nm
long pass filter was used to maximise the signal and achieved a quantum
contrast zhu2021high  of 1800 and collection efficiencies of 10.9% and 9.6%
for each SPAD.

Once aligned, the long pass filter was replaced with a 790 nm bandpass filter,
$F_{2}$ (Thorlabs FBH790-10) with a full width at half maximum (FWHM) of 10
nm, narrowing the spectrum in the frequency domain in order to broaden the
width in the time domain. The photon pairs were imaged using another lens
($L_{7}$) with $f_{2}~{}=60$ mm. The SM fibers were connected to a SM FBS
which was connected to two SPADs (Excelitas SPCM-NIR), each recording singles
counts. The SPADs are enhanced for detecting in the near infrared (NIR)
region, with a peak photon detection efficiency (PDE) of 70% for 780 nm
wavelengths. The time tags of the photon arrival time were recorded using a
PicoQuant TimeHarp 200 allowing coincidences to be established between the two
detectors within a 1 ns window.

The second arm contains a retroreflector prism (P) mounted on a motorised
translation stage (CONEX-MFACC), which was scanned to find the dip in
coincidences counts. When the location of the dip was found, fiber
polarization controllers were rotated to match the polarization of the two
photons by minimising the dip in coincidence counts at the dip location. Once
the coincidences counts have been maximised outside the dip and minimised
inside the dip, the coincidence data as a function of stage position can be
recorded.

Figure 2:  (a) The output power as a function of pump diode power, with a
linear fit showing a slope efficiency of 12.4%. Self-starting Kerr-lens-
modelocking occurs for pump powers greater than 1005 mW as indicated by the
blue shaded region. (b) Modelocked spectra recorded for pump powers of 1100 mW
with a central wavelength of 791 nm and FWHM of 8.1 nm. (c) Radio-frequency
spectrum recorded at the instrument-limited resolution bandwidth of 15 kHz.
The repetition rate is centered at 1.02 GHz. (d)Detail of the fundamental peak
of the radio-frequency spectrum shown in (c).

The characteristics of the diode pumped three-element Kerr-lens-modelocked
Ti:sapphire cavity are shown in Fig. 2. The output power as a function of the
pump power is shown in Fig. 2 (a) and a linear fit implied the slope
efficiency of the laser was 12.4%. Self-starting Kerr-lens-modelocking occurs
for pump powers greater than 1005 mW, as indicated by the shaded region. The
normalised modelocked spectrum is shown in Fig. 2 (b) and has a central
wavelength of 791 nm and a FWHM of 8.1 nm. Finally, the radio-frequency
spectrum was recorded with an instrument-limited resolution bandwidth of 15
kHz and is shown in Fig. 2 (c). A zoomed in view of the radio-frequency
spectrum is shown in Fig. 2 (d) and the repetition rate of the modelocked
laser is found to be 1.02 GHz.

##  III Results

Using this custom built Ti:sapphire modelocked laser (see ostapenko2022three ;
ostapenko2023design  for more information), we were able to observe a HOM dip
with a visibility of 81.8%, as shown in Fig. 3. Coincidences were recorded for
three minutes for every measurement and the FWHM of the dip was $110~{}\mu$m.
In order to maximise the visibility of the HOM dip, the photons must be
indistinguishable when interfering at the FBS. That is to say, that if the two
photons have the same polarization and spectral properties and arrive at the
FBS at the same time, the visibility will approach 100%. To further improve
the observed visibility, we would require more signal power to see changes to
the coincidences inside the dip in real time.

Figure 3:  HOM interference between signal and idler photons. A HOM dip is
observed with a visibility of 81.8% and a FWHM of $110~{}\mu m$. The error
bars are calculated using Poisson statistics. The solid line is a Gaussian
fit.

##  IV Conclusion

In this work, we generate single photon pairs via SPDC using a compact, diode-
pumped three-element Kerr-lens-modelocked Ti:sapphire cavity with a 1-GHz
repetition rate ostapenko2022three ; ostapenko2023design . We verify the
presence of indistinguishable photon pairs using HOM interferometry by
recording coincidence counts with SPADs and observe a dip in coincidences with
a visibility of 81.8%. These results show that we were able to create a
quantum source of indistinguishable photon pairs using a simple and compact
GHz-rate laser, opening new opportunities for GHz rate SPDC generation and
detection in systems with low size, weight and power.

We could achieve GHz-rate SPDC generation by removing the upconversion process
described above, down-converting the 790 nm beam and detecting single photon
pairs using superconducting nanowire single-photon detectors (SNSPDs). These
detectors have a higher quantum efficiency and detection rate compared to
SPADs, with quantum efficiencies of 98% reported for detection at 1550 nm
wavelengths reddy2020superconducting  compared to a 70% quantum efficiency at
780 nm wavelengths for the SPADs used in this experiment. The elimination of
the up-conversion process would reduce the loss in laser power and increase
the probability of producing a photon pair in the SPDC process.

In conclusion, we have generated indistinguishable photon pairs using a GHz
rate diode-pumped laser, which is compact and low cost in comparison to
typical ultrafast lasers used in SPDC to date. These characteristics make such
a laser source a prime candidate in applications such as satellite QKD, where
the size, weight and power of the laser are critical.

## Funding

Science and Technology Facilities Council (ST/S505407/1, ST/V00040/1);
Engineering and Physical Sciences Research Council (EP/S001638/1,
EP/T00097X/1, EP/P005446/1).

## Disclosures

The authors declare no conflicts of interest.

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# Note on a note of Goldston and Suriajaya

J.B. Friedlander∗ and  H. Iwaniec

∗ Supported in part by NSERC grant A5123

Abstract: We show that the assumption of a weak form of the Hardy-Littlewood
conjecture on the Goldbach problem suffices to disprove the possible existence
of exceptional zeros of Dirichlet $L$-functions. This strengthens a result of
the authors named in the title.

##  1\. Introduction

A fundamental problem in analytic number theory is that of establishing zero-
free regions for Dirichlet $L$-functions. In case the corresponding character
$\chi\mathop{\rm{mod}}q$ is complex, or alternatively, in the case of complex
zeros $\rho=\beta+i\gamma$ with $\gamma\neq 0$, one has long known how to
produce zero-free regions of the type

(1.1) $\sigma\geqslant 1-c/\log q(|t|+2)$

where $s=\sigma+it$. In the case both $\chi$ and $s$ are real much less is
known, nothing more recent than a famous “ineffective” estimate of Siegel
which replaces $\log q$ by $q^{\varepsilon}$. This exponentially weaker result
has been a serious impediment to progress in many basic questions.

In the absence of a solution to the problem of these exceptional zeros there
are some useful results showing their rarity. There have also been attempts to
relate the question to other very difficult problems. One famous example, by
now folklore, shows that the non-existence of such zeros would follow from
improvements seductively small in the Brun-Titchmarsh theorem which gives
uniform upper bounds for the number of primes in an arithmetic progression.

In more recent years some results have been obtained by showing how relatively
good bounds for exceptional zeros would follow from assumptions about the
seemingly less directly related Goldbach conjecture. The latter famous
statement predicts that every even integer exceeding two can be written as the
sum of two primes. Hardy and Littlewood [HL] put forth a conjectured
asymptotic formula for the number of representations of $n$ as the sum of two
primes. Following the normal practice in the subject, we find it simpler to
consider a weighted sum over the repsentations involving the von Mangoldt
function, one which leads to an entirely equivalent conjecture. Let

(1.2) $G(n)=\sum_{\begin{subarray}{c}m_{1}+m_{2}=n\\\ 2\nmid
m_{1}m_{2}\end{subarray}}\Lambda(m_{1})\Lambda(m_{2}).$

The Hardy-Littlewood conjecture predicts that, for $n$ even, we have
$G(n)\sim{\mathfrak{S}}(n)n$ where ${\mathfrak{S}}(n)$ is a certain positive
product over the primes, defined in (6.2) to (6.4) and easily large enough to
imply Goldbach for all large $n$.

A rather weakened, but still formidable, form of the Hardy-Littlewood
conjecture which features in this (and in earlier) work is as follows.

Weak Hardy-Littlewood-Goldbach Conjecture: For all sufficiently large even $n$
we have

(1.3) $\delta{\mathfrak{S}}(n)n<G(n)<(2-\delta){\mathfrak{S}}(n)n,$

for some fixed $0<\delta<1$.

Despite having a slightly different name, this is the same conjecture employed
in [GS]. We are going to prove the following result.

######  THEOREM 1.

Assume that the Weak Hardy-Littlewood-Goldbach Conjecture (1.3) holds for all
sufficiently large even $n$. Then there are no zeros of any Dirichlet
$L$-function in the region (1.1) with a positive constant $c$ which is now
allowed to depend on $\delta$.

Of course, as remarked above, we are allowed to restrict the proof to the case
of real zeros of $L$-functions with real characters. We can also assume that
the modulus $q$ is large since if there were only a finite number of
exceptional zeros we could reach a region of type (1.1) simply by adjusting
the constant $c$ (since $L(1,\chi)>0$).

Precursors of this work include [Fe], [BHMS], [BH], [Ji] and [GS]. A
discussion of their results can be found in [GS]. In those works the results
led to an interval of width $c/(\log q)^{2}$ rather than $c/\log q$. Aside
from an only slightly different approach, our main innovation is the use of a
theorem of Bombieri, described in Section 4, which allows us to incorporate
the zero repulsion principle used, for example, in connection with Linnik’s
theorem on the least prime in an arithmetic progression.

Acknowledgement We were completely motivated to consider this problem by the
paper [GS] of Goldston and Suriajaya and by the beautiful lecture of Dan
Goldston delivered to the web seminar of the American Institute of Mathematics
on May 4, 2021. We are also grateful to the Institute for having provided this
venue.

##  2\. A Goldbach number generating series

We let $N$ and $q$ be given positive integers. We consider the sum

(2.1) $S(q)=\sum_{n\equiv 0(\mathop{\rm{mod}}q)}G(n)e^{-n/N}$

where $G(n)$ is given by (1.2). We are going to estimate $S(q)$ in two
different ways and then compare the results.

##  3\. Zeros of $L$-functions and an explicit formula

By the orthogonality property for the characters we write

$\displaystyle S(q)$ $\displaystyle=\sum_{\begin{subarray}{c}m_{1}+m_{2}\equiv
0(\mathop{\rm{mod}}q)\\\ 2\nmid
m_{1}m_{2}\end{subarray}}\Lambda(m_{1})\Lambda(m_{2})e^{-(m_{1}+m_{2})/N}$
$\displaystyle=\frac{1}{\varphi(q)}\sum_{\chi(\mathop{\rm{mod}}q)}\chi(-1)\sum_{m}\Bigl{(}\chi(m)\Lambda(m)e^{-m/N}\Bigr{)}^{2}+O\Bigl{(}N(\log
q)^{2}\Bigr{)}\ $

where the error term accounts for a trivial estimation of the contribution to
the sum from terms $m_{1}$, $m_{2}$ with $(m_{1}m_{2},2q)\neq 1$.

Let $N\geqslant q$. The principal character contributes to $S(q)$ an amount

(3.1)
$S_{0}(q)=\frac{1}{\varphi(q)}\Bigl{(}\sum_{(m,q)=1}\Lambda(m)e^{-m/N}\Bigr{)}^{2}=\frac{N^{2}}{\varphi(q)}\Bigl{(}1+O\bigl{(}\frac{1}{\log
N}\bigr{)}\Bigr{)}.$

The exceptional character $\chi_{1}(\mathop{\rm{mod}}\,q)$ contributes

(3.2) $S_{1}(q)=\frac{\chi_{1}(-1)}{\varphi(q)}P(\chi_{1})^{2}$

where for any $\chi$ we have

(3.3) $P(\chi)=\sum_{m}\chi(m)\Lambda(m)e^{-m/N}.$

We have the trivial bound $P(\chi)\ll N$. Hence, all the other characters
$\chi\neq\chi_{0},\chi_{1}$, contribute at most

(3.4)
$S_{\infty}(q)\ll\frac{N}{\varphi(q)}\sum_{\chi\neq\chi_{0},\chi_{1}}|P(\chi)|.$

We evaluate $P(\chi)$ for each $\chi\neq\chi_{0}$ as follows:

(3.5) $\displaystyle P(\chi)$ $\displaystyle=\frac{1}{2\pi
i}\int_{(2)}\Gamma(s)N^{s}\frac{-L^{\prime}}{L}(s)ds$
$\displaystyle=-\sum_{\rho}\Gamma(\rho)N^{\rho}+O\Bigl{(}N^{\frac{1}{2}}(\log
N)^{2}\Bigr{)},$

on moving the line of integration to ${\rm Re\,s}=\frac{1}{2}$. Here,
$\rho=\beta+i\gamma$ runs over the zeros of $L(s,\chi)$ with
$\frac{1}{2}\leqslant\beta<1$, giving the explicit formula for the sum
$P(\chi)$ in (3.3).

##  4\. Linnik’s zero repulsion à la Bombieri

We have the following result.

######  PROPOSITION 4.1.

Suppose there is a real character $\chi_{1}(\mathop{\rm{mod}}q)$ with a real
zero $\beta_{1}$ of $L(s,\chi_{1})$ in the segment

(4.1) $1-\frac{c}{\log q}\leqslant\beta_{1}<1$

where $c$ is a small positive constant. Let $N(\alpha,T)$ denote the number of
zeros, $\rho=\beta+i\gamma$, $\rho\neq\beta_{1}$ of $L(s,\chi)$ for all
$\chi(\mathop{\rm{mod}}q)$, counted with multiplicity, in the region

(4.2) $\alpha\leqslant\beta<1,\,\,|\gamma|\leqslant T.$

Then, for $\frac{1}{2}\leqslant\alpha\leqslant 1,T\geqslant 2$

(4.3) $N(\alpha,T)\ll(1-\beta_{1})(\log q)q^{b(1-\alpha)}T^{3}$

where $b$ and the implied constant are absolute.

This proposition follows from Théorème 14 of Bombieri [B] which states the
following.

######  PROPOSITION 4.2.

For $\frac{1}{2}\leqslant\alpha\leqslant 1$ and $T\geqslant 2$, we have

$\sum_{q\leqslant
T}\,\sideset{}{{}^{*}}{\sum}_{\chi(\mathop{\rm{mod}}q)}N(\alpha,T,\chi)\ll(1-\beta_{1})(\log
T)T^{b(1-\alpha)}$

where $\chi$ runs over primitive characters and $N(\alpha,T,\chi)$ is the
number of zeros $\rho=\beta+i\gamma$, of $L(s,\chi)$ counted with
multiplicity, in the region (4.2), apart from one real and simple zero
$\beta_{1}\geqslant 1-c/\log T$.

To deduce Proposition 4.1 we first consider the case $T\leqslant q$. Now, our
sum $N(\alpha,T)$ satisfies

$\displaystyle N(\alpha,T)$ $\displaystyle\leqslant
N(\alpha,q)\leqslant\sum_{\chi(\mathop{\rm{mod}}q)}N(\alpha,q,\chi)=\sum_{k|q}\sideset{}{{}^{*}}{\sum}_{\chi(\mathop{\rm{mod}}k)}N(\alpha,q,\chi)$
$\displaystyle\leqslant\sum_{k\leqslant
q}\,\sideset{}{{}^{*}}{\sum}_{\chi(\mathop{\rm{mod}}k)}N(\alpha,q,\chi)\ll(1-\beta_{1})(\log
q)q^{b(1-\alpha)},$

giving the result in this case.

On the other hand, if $T>q$ the result follows from the classical bound
$N(T,\chi)\ll T\log qT$.

We remark that the factor $T^{3}$ in (4.3) is wasteful but simplifies the
proof and is unimportant for our application.

##  5\. First estimation of $S(q)$

We are now ready to complete the estimation of $S(q)$ by means of Dirichlet
characters. We input the result of Proposition 4.1. We derive by partial
summation that

(5.1)
$\sum_{\chi(\mathop{\rm{mod}}q)}\,\sum_{\rho\neq\beta_{1}}|\Gamma(\rho)N^{\rho}|\ll
N(1-\beta_{1})\log q,$

provided $N\geqslant q^{b+1}$. Hence, by (3.4) and (3.5), we have

(5.2) $S_{\infty}(q)\ll\frac{N^{2}}{\varphi(q)}(1-\beta_{1})\log q.$

On the other hand, taking only the exceptional character rather than the
others in (5.1), we get

(5.3) $P(\chi_{1})=-\Gamma(\beta_{1})N^{\beta_{1}}+O\bigl{(}N(1-\beta_{1})\log
q\bigr{)}$

and so

(5.4)
$P(\chi_{1})^{2}=\Gamma(\beta_{1})^{2}N^{2\beta_{1}}+O\bigl{(}N^{2}(1-\beta_{1})\log
q\bigr{)}.$

Finally, adding the above estimates we obtain

(5.5) $\displaystyle S(q)$
$\displaystyle=S_{0}(q)+\frac{\chi_{1}(-1)}{\varphi(q)}N^{2\beta_{1}}+O\Bigl{(}\frac{N^{2}}{\varphi(q)}(1-\beta_{1})\log
q\Bigr{)}$
$\displaystyle=\bigl{(}1+\chi_{1}(-1)+\varepsilon(q,N)\bigr{)}\frac{N^{2}}{\varphi(q)},$

where

(5.6) $\varepsilon(q,N)\ll(1-\beta_{1})\log N+(\log N)^{-1}$

if $N\geqslant q^{b+1}$. The implied constant is absolute.

##  6\. A model of $S(q)$

In this section we consider the a model for $S(q)$ to be given by (2.1).
Although our computations in this section, as for those earlier, are
unconditional, the closeness of the model to the sum $S(q)$ is not. That will
rest on the assumption in the next section.

The Hardy-Littlewood conjecture for the Goldbach problem asserts that, for $n$
even,

(6.1) $G(n)\sim{\mathfrak{S}}(n)n,$

where

(6.2) ${\mathfrak{S}}(n)=2CH(n),$

and, in turn, $C$ is the positive absolute constant

(6.3) $C=\prod_{p>2}\Bigl{(}1-\frac{1}{(p-1)^{2}}\Bigr{)},$

and $H(n)$ is the multiplicative function

(6.4) $H(n)=\prod_{\begin{subarray}{c}p|n\\\
p>2\end{subarray}}\Bigl{(}1+\frac{1}{(p-2)}\Bigr{)}.$

######  LEMMA 6.1.

We have

(6.5) $\sum_{\begin{subarray}{c}n\equiv 0(\mathop{\rm{mod}}q)\\\ n\,\,{\rm
even}\end{subarray}}{\mathfrak{S}}(n)ne^{-n/N}=\frac{N^{2}}{\varphi(q)}\Bigl{(}1+O\Bigl{(}\frac{q}{N}\Bigr{)}^{\frac{1}{2}}\Bigr{)}.$

###### Proof.

Pulling out the constant factor $2C$ we evaluate the sum

(6.6) $\sum_{\begin{subarray}{c}n\equiv 0(\mathop{\rm{mod}}q)\\\ n\,\,{\rm
even}\end{subarray}}H(n)ne^{-n/N}=\frac{N}{2\pi
i}\int_{(2)}\Gamma(s+1)N^{s}Z_{q}(s)ds$

where

$\ Z_{q}(s)=\sum_{n\equiv 0(\mathop{\rm{mod}}k)}H(n)n^{-s}$

and $k=[2,q]=2q/(2,q)$. We have

$\displaystyle Z_{q}(s)$
$\displaystyle=H(k)k^{-s}\sum_{\ell=1}^{\infty}\prod_{\begin{subarray}{c}p|\ell\\\
p\nmid k\end{subarray}}\Bigl{(}1+\frac{1}{p-2}\Bigr{)}\ell^{-s}$
$\displaystyle=H(k)k^{-s}\prod_{p|k}\Bigl{(}1-\frac{1}{p^{s}}\Bigr{)}\prod_{p\nmid
k}\Bigl{(}1+\Bigl{(}1+\frac{1}{p-2}\Bigr{)}\frac{1}{p^{s}}\Bigl{(}1-\frac{1}{p^{s}}\Bigr{)}^{-1}\Bigr{)}$
$\displaystyle=H(k)k^{-s}\zeta(s)\prod_{p\nmid
k}\Bigl{(}1+\frac{1}{p^{s}(p-2)}\Bigr{)}.$

This has a simple pole at $s=1$ with residue

(6.7) ${\rm res}_{s=1}Z_{q}(s)=\frac{H(k)}{k}\prod_{p\nmid
k}\Bigl{(}1+\frac{1}{p(p-2)}\Bigr{)}=\frac{1}{2C\varphi(q)}.$

Moving the line of integration in (6.6) to ${\rm Re\,s}=\frac{1}{2}$, we
obtain (6.5).

∎

##  7\. Completion

In this section we begin by completing our second estimation for $S(q)$ by
comparing its model. In this we need to draw on an unproved assumption. We
assume the following.

Weak Hardy-Littlewood-Goldbach Conjecture: For all sufficiently large even $n$
we have

(7.1) $\delta{\mathfrak{S}}(n)n<G(n)<(2-\delta){\mathfrak{S}}(n)n,$

where $0<\delta<1$ is fixed.

From (7.1) and Lemma 6.1 we see that, with $N\geqslant q^{2}$,

(7.2) $\delta\frac{N^{2}}{\varphi(q)}\Bigl{(}1+O\Bigl{(}\frac{1}{\log
N}\Bigr{)}\Bigr{)}<S(q)<(2-\delta)\frac{N^{2}}{\varphi(q)}\Bigl{(}1+O\Bigl{(}\frac{1}{\log
N}\Bigr{)}\Bigr{)}.$

We now combine our two estimations for $S(q)$. By (7.2) and (5.5) we have

(7.3) $\delta<1+\chi_{1}(-1)+\varepsilon(q,N)<2-\delta$

if $N\geqslant q^{b+1}$, where (a now slightly different) $\varepsilon(q,N)$
still satisfies (5.6). Taking the constant $c$ in (4.1) to be sufficiently
small, we see from (5.6) that

(7.4) $|\varepsilon(q,N)|\leqslant\tfrac{\delta}{2}.$

Since $1+\chi_{1}(-1)=0$ or $=2$, this is not possible.

Conclusion: The assumption (7.1) with some positive $\delta$ implies that
there is a real zero-free interval $s\geqslant 1-c(\delta)/\log q$, (and hence
a zero-free region in the plane $\sigma\geqslant 1-c(\delta)/\log q(|t|+2)$
for all $L(s,\chi)$ to all moduli $q$). In the case that $\chi_{1}(-1)=1$ only
the upper bound in (7.1) is required and in the other case, that
$\chi_{1}(-1)=-1$, only the lower bound is required.

## References

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  * [BH] G. Bhowmik and K. Halupczok, Conditional Bounds on Siegel Zeros, arXiv:2010.01308 [math.NT].
  * [B] E. Bombieri, Le Grand Crible dans la Théorie Analytique des Nombres, Astérisque 18, Soc. Math. France, 2ieme ed., (Paris) 1987/1974.
  * [Fe] J.H. Fei, An application of the Hardy-Littlewood conjecture, J. Number Theory 168, (2016), 39–44.
  * [GS] D.A. Goldston and A.I. Suriajaya, Note on the Goldbach conjecture and Landau–Siegel zeros, arXiv:2104.09407v1 [math.NT].
  * [HL] G.H. Hardy and J.E. Littlewood, Some problems of ‘Partitio Numerorum’: III. On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1–70.
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Department of Mathematics, University of Toronto

Toronto, Ontario M5S 2E4, Canada

Department of Mathematics, Rutgers University

Piscataway, NJ 08903, USA