E-mails: {jjkobak, bjrichman, lewashington<EMAIL_ADDRESS>
# Continuous Human Activity Recognition using a MIMO Radar for Transitional
Motion Analysis

John Kobak  School of Engineering, Widener University, Chester, PA 19013, USA
Bennett J. Richman  School of Engineering, Widener University, Chester, PA
19013, USA  LaJuan Washington Jr  School of Engineering, Widener University,
Chester, PA 19013, USA  Syed A. Hamza  School of Engineering, Widener
University, Chester, PA 19013, USA

###### Abstract

The prompt and accurate recognition of Continuous Human Activity (CHAR) is
critical in identifying and responding to health events, particularly fall
risk assessment. In this paper, we examine a multi-antenna radar system that
can process radar data returns for multiple individuals in an indoor setting,
enabling CHAR for multiple subjects. This requires combining spatial and
temporal signal processing techniques through micro-Doppler (MD) analysis and
high-resolution receive beamforming. We employ delay and sum beamforming to
capture MD signatures at three different directions of observation. As MD
images may contain multiple activities, we segment the three MD signatures
using an STA/LTA algorithm. MD segmentation ensures that each MD segment
represents a single human motion activity. Finally, the segmented MD image is
resized and processed through a convolutional neural network (CNN) to classify
motion against each MD segment.

##  1 Introduction

The use of contactless technology for detecting human motions has become
increasingly popular due to its non-invasive nature as it eliminates the need
for users to wear specific tracking devices [1, 2, 3]. Radar systems are
particularly effective in this regard, as they provide reliable non-contact
monitoring that is privacy-preserving and not affected by lighting conditions.
Active radio frequency (RF) sensing, in particular, allows for 4D imaging
capabilities by measuring the scatterer’s velocity in addition to range and 2D
angular localization. This is a major advantage over visual-based systems,
which require additional pre-processing and filtering operations to accurately
detect small movements [4, 5]. While a high-resolution imaging radar ensures
privacy by generating silhouette-type portraits that reveal minimal
identifiable information, unlike camera-based systems. Radar-based CHAR has
diverse applications such as detecting significant events for automated
surveillance, analyzing daily activities and behaviors, recognizing
abnormalities in gait, monitoring health in care facilities and rehabilitation
services, and promoting independent living for the elderly. [6, 7, 8, 9, 10,
11, 5].

The primary objective of this research is to detect the real-time daily
activities of one or multiple individuals residing in a shared dwelling.
Utilizing radar technology to achieve effective CHAR presents a significant
challenge, as it relies heavily on a series of pre-processing steps that must
isolate both the individuals and their corresponding activities
simultaneously. The proposed approach in this paper utilizes beamforming to
filter the received data and achieve spatial filtering, effectively isolating
the motion of a single individual in the field of view. As a result of human
motion, the radar return undergoes a frequency shift due to the Doppler
effect. However, since human movement involves the motion of various body
parts, there are often additional movements and rotations in different parts
of the body besides the torso’s movement. For example, when a person walks,
their arms swing naturally. These micro-scale movements result in additional
Doppler shifts, known as micro-Doppler (MD) effects, which aid in identifying
the motion type. After isolating an individual in a certain angular region, in
the field of view, short-time Fourier transform is applied to the output of
the beamformer to obtain a MD image. However, due to the limited angular
resolution of the beamformer, the actions performed by individuals outside of
the main beam may not be entirely separable, either due to their proximity to
the other individual or leakage associated with the beamformer sidelobes.
Hence, MD images might not entirely be representative of the activity of the
individual in the main beam but also has remnants of the activities performed
by other individuals at different locations. At this stage, it is possible
that the MD image could include a series of activities, and the goal is to
separate the series of activities executed by the individual present within
the beamformer’s main lobe. We apply STA/LTA (short-time-average/long-time-
average trigger algorithm) in order to segment the MD image into sub-images,
such that each sub-image comprise exclusively of a single motion. The STA/LTA
algorithm performs event detection by utilizing two windows that operate on
the envelopes of the MD image. The envelopes are extracted after smoothing out
the modulated MD image, which facilitates the envelope detection algorithm
employing the percentile technique. The STA/LTA algorithm’s task of detecting
the activity of a specific individual within the main beam is challenging due
to the presence of residual activities from other individuals that were not
intended to be captured in the MD image obtained after spatial filtering. In
the final step, the sub-images are padded to a uniform size and fed into a
convolutional neural network (CNN) to classify and recognize the specific
human activity being performed in the image sequence. It is noted that padding
is necessary for resizing the sub-images to achieve consistent input
dimensions, as the sub-dimensions of images may vary depending on the
activity. Two experimental studies were conducted to collect data, one
involved a single subject while the other involved two subjects using
beamforming. A total of 800 trials were performed, with various activities
carried out by a maximum of two individuals.

The paper’s organization is as follows: Section 2 details the signal model
discussing the methodology to process ADC data returns to convert them into
visible MD spectrograms. Section 3 details the radar parameters and discusses
the data collection process. Section 4 describes the procedure of cleaning the
MD images and separating the individual events within a MD image. Section 5
outlines the structure of the proposed CNN for one person, and then
extrapolate that to include a combination of right, left, and nominal
spectrogram images. Finally, Section 6 reveals the results.

##  2 Radar return signal analysis

The complex valued raw data matrix $\mathbf{s}(n,m)\in C^{N*M}$ of the
frequency-modulated continuous wave (FMCW) radar is obtained through spatially
processing the radar returns by an $M$ element uniformly spaced antenna array.
The data is collected over $N$ temporal sampling instances. The receiver array
vector $\mathbf{s}(m)\in C^{M}$ at time instant $n$ corresponds to the
$n_{th}$ row of $\mathbf{s}(n,m)$ and is given by,

$\mathbf{s}(m)=\sum_{l=1}^{L}\alpha_{l}\mathbf{a}(\theta_{l})+\mathbf{v}(m),$
(1)

where, $\mathbf{a}({\theta_{l}})$ $\in\mathbb{C}^{M}$ is the steering vector
corresponding to the azimuth direction $\theta_{l}$ of the scatterer, and is
defined as follows,

$\mathbf{a}({\theta_{l}})=[1\,\,\,e^{j(2\pi/\lambda)dcos(\theta_{l})}\,.\,.\,.\,e^{j(2\pi/\lambda)d(M-1)cos(\theta_{l})}]^{T}.$
(2)

Here, $d$ is the inter-element spacing and $\alpha_{l}$ $\in\mathbb{C}$ is the
complex amplitude of the radar return. The additive Gaussian noise
$\mathbf{v}(m)$ $\in\mathbb{C}^{M}$ has variance $\sigma_{v}^{2}$. The
elements of the received data vector $\mathbf{s}(m)$ are combined linearly by
the $M$-sensor beamformer that strives to spatially filter the reflections
from all other directions except the signal in the direction of beamformer
look angle $\theta_{k}$. The spatially filtered signal vector
$\mathbf{x}({\theta_{k}})$ $\in\mathbb{C}^{N}$ after beamforming is given by,

$\mathbf{x}({\theta_{k}})=\mathbf{s}(n,m)\mathbf{w}^{H}({\theta_{k}}),$ (3)

where $\mathbf{w}({\theta_{k}})=\mathbf{a}^{H}({\theta_{k}})$ are the complex
beamformer weights pointing towards $\theta_{k}$.

The spatially filtered signal vector $\mathbf{x}({\theta_{k}})$ is reshaped
into a two-dimensional matrix, $\mathbf{x}_{\theta_{k}}(p,q)$. This is
achieved by segmenting the $N$ dimensional vector $\mathbf{x}({\theta_{k}})$,
such that, the $P$ samples collected within a pulse repetition interval (PRI)
are stacked into a $P$ dimensional column. There are $Q$ such columns within
$\mathbf{x}_{\theta}(p,q)$ where $Q=N/P$ is the number of PRIs processed
within the observation time $N$. The range-map, $\mathbf{r}_{\theta_{k}}(p,q)$
is obtained by applying the column-wise Discrete Fourier Transform (DFT)
operation which is given by,

$\mathbf{r}_{\theta_{k}}(l,q)=\sum_{p=0}^{P-1}\mathbf{x}_{\theta_{k}}(p,q)e^{-j(2\pi
lp/N)}$ (4)

We observe the data in the time-frequency (TF) domain after localizing the
motion in azimuth and range bins of interest. The spectrogram is used as the
TF signal representation, showing the variation of the signal power as a
function of time $n$ and frequency $k$. The spectrogram of a periodic version
of a discrete signal $\mathbf{v}_{\theta_{k}}(n)$, is given by [12, 13, 14],

$\mathbf{d}_{\theta_{k}}(n,k)=|\sum_{m=0}^{H-1}\mathbf{h}(m)\mathbf{v}_{\theta_{k}}(n-m)e^{-j(2\pi
km/H)}|^{2},$ (5)

where
$\mathbf{v}_{\theta_{k}}=\sum_{l=r_{l}}^{r_{u}}\mathbf{r}_{\theta_{k}}(l,q)$
is obtained by collapsing the range dimension beginning from lower range bin
$r_{l}$ to highest range bin $r_{h}$. Tapering window $\mathbf{h}$ of length
$H$ is applied to reduce the sidelobes. The spectrograms reveal the different
velocities, accelerations, and higher order moments which cannot be easily
modeled or assumed to follow specific non-stationary structures [15, 16]. We
observe the motion of two persons performing different activities in close
proximity to each other at different azimuth angles. We aim to correctly pair
the activity to the corresponding azimuth angle. This is achieved by jointly
processing the spectrograms, $\mathbf{v}_{\theta_{1}}(n,k)$ and
$\mathbf{v}_{\theta_{2}}(n,k)$ which are respectively localized at azimuth
angles $\theta_{1}$ and $\theta_{2}$. It is clear that the actions of multiple
persons is hard to be distinguished in azimuth by only using a single antenna.

##  3 Data Collection

To ensure the representation of the average human body type, we performed each
transitional activity using six subjects with different heights, genders and
body compositions. The number of trials conducted for each activity is
outlined in Table 1. Please note that these classes arise after performing
image segmentation on extended MD images. For single person activity, the
extended MD images were captured for 12s with different combinations of three
human motions selected from five activities namely, walking forward, walking
backward, sitting, standing, and bending motions. For the two-person data,
trials were conducted for walking forward and walking back at $\pm 30$°,
resulting in four additional classes. The radar system used had a bandwidth of
4 GHz and operated at 77 GHz. The two participants performed activities while
facing the radar system with radial angles of +30°and -30°, respectively. The
radar systems were positioned at an average distance of 2 meters from the
participants, and the RF sensor’s output transmission power was set to 40 mW.

  * •

PRI is set to 1 ms, and each data example is observed over the time period of
12 s, resulting in $Q=4000$ slow time samples.

  * •

ADC sampling rate is 512 ksps, rendering 512 fast time samples per PRI,
resultantly the length of data vector is $N=6156000$.

  * •

The received data $\mathbf{s}(n,m)\in C^{N*M}$, is collected through $M=4$
element receive array, with an inter-element spacing of $\lambda/2$ ($\lambda$
is the wavelength corresponding to the operating frequency), therefore the
dimensionality of received raw data matrix is $6156000\times 4$.

  * •

Beamforming is performed on the raw data matrix, resulting in a spatially
filtered $\mathbf{x}({\theta_{k}})$ vector of dimensions $3276800\times 1$.
Two such vectors are generated in the directions of each motion ${\theta_{1}}$
and ${\theta_{2}}$.

  * •

Each vector $\mathbf{x}({\theta_{k}})$ is reshaped into a $512\times 12000$
matrix. After applying columnwise DFT, and identifying the range bins of
interest, the corresponding rows are summed together, resulting in
$\mathbf{v}_{\theta_{k}}=\sum_{l=r_{l}}^{r_{u}}\mathbf{r}_{\theta_{k}}(l,q)$,
which is of dimensions $12000\times 1$.

  * •

A combined spectrogram and two spectrograms after beamforming
$\mathbf{d}_{\theta_{1}}$ and $\mathbf{d}_{\theta_{2}}$, each of dimensions
$128\times 128$ is obtained, where the window length is 128.

Examples |  Forward |  Backward |  Bending |  Standing |  Sitting down |  Forward +30° |  Forward -30° |  Backward +30° |  Backward -30°
---|---|---|---|---|---|---|---|---|---
Total |  771 |  498 |  120 |  71 |  131 |  13 |  41 |  31 |  40
Table 1:  Number of trials for segmented activity

##  4 Exyended Micro-Doppler Segmentation

Figure 1: Flowchart for the proposed approach

Figure 1 depicts the complete processing chain, beginning with radar data
collection and culminating in the classification of a specific human activity
type. Once the extended MD images are acquired, after beamforming[17], the
envelope and event detection algorithms are applied to locate the start and
end times of possible events within the duration of the extended MD image.
These timestamps are then used to identify and crop the individual events
within the extended MD image. Next, the cropped images are resized to a
resolution of $600\times 600$ by zero-padding, enabling all of them to be
consistent with the input dimensions of the CNN. Event detection is
accomplished using the STA/LTA algorithm, which is elaborated on below.

###  4.1 STA/LTA algorithm for Event Detection

The short-time-average/long-time-average STA/LTA trigger algorithm is commonly
utilized in weak-motion applications where the detection of events is most
desired. With extended MD returns providing TF characteristics of reflected
Doppler signals, the STA/LTA algorithm can identify and extract the desired
features by tracking the start and end times of events. The STA/LTA algorithm
is applied to the envelope of the extended MD which is obtained using an
envelope detection algorithm.

Figure 2: Pre-Processing Stage Implementation

Left: Envelope Detection, Center: Event Detection, Right: Zero-Padded Image

Envelope detection is a technique used to extract the envelope of the
modulated extended MD signal, which essentially captures the highest and
lowest amplitude variations at all time instances. When a extended MD image is
utilized as the primary signal, the Van Dorp and Groen percentile technique
can be employed to reduce the radar return into a single array. This
percentile technique calculates the cumulative amplitude distribution for each
temporal slice[18]. The common approach to implementing this technique with a
extended MD return is to find the upper $u_{i}$, central $c_{i}$, and lower
$l_{i}$ envelopes through percentile multiplication with the sum of
intensities with time $I(n)$ as follows,

$u_{i}(n)=0.97*I(n),$ (6) $c_{i}(n)=0.50*I(n),$ (7)

and

$l_{i}(n)=0.03*I(n).$ (8)

Because background noise cannot be completely filtered out of the extended MD
during the smoothing process, shown in Figure 2 Left, we propose utilizing two
central envelopes, starting from opposite ends of the extended MD image, and
averaging across them in time to obtain the ‘real’ central envelope.

After obtaining the central envelope, the STA/LTA ratio is calculated
continuously at each time sample $n$ for every $j$th pixel along the central
envelope $c_{i}$ as $R={\frac{STA(n)}{LTA(n)}}$[19], where

$STA(n)=\frac{1}{N_{1}}\sum_{j=n+1}^{n+N_{1}}{c_{i}}(j),$ (9)

and

$LTA(n)=\frac{1}{N_{2}}\sum_{j=n-N_{2}}^{n}{c_{i}}(j).$ (10)

$N_{1}$ and $N_{2}$ are short/long window lengths, respectively. The start
time of an event is determined when $STA(n)>\sigma_{1}$ and $R>\sigma_{2}$ and
ends when $STA(n)<\sigma_{3}$ and $R<\sigma_{2}$ where
$\sigma_{1},\sigma_{2},\sigma_{3}$ are pre-defined detection thresholds. Note
that we have considered non-overlapping STA and LTA windows.

Ideally, the triggered event should include all phases, but for the majority
of these weak events, the algorithm does not trigger at the beginning because
the threshold for ignition of the start time is reached a couple of pixels
late. These detection thresholds are set based on the maximum frequency pixel
of non-motion observation. To counteract this side effect, we propose an
appropriate pre-event time (PEM) selection that ensures the correct start time
of the event [20]. This same concept of incorrect ignition due to threshold
setting also occurs during detriggering of an event. Because of this, we also
set a post-event time (PET) parameter. Optimal PEM and PET durations depend
mostly on the application. For the underlying scenario, the extended MD image
spans over twelve-second, PEM=PET=${\frac{SizeofEvent}{20}}$.

It’s worth noting that energy leakage due to beamforming on radar returns at
varying angles can potentially lead to issues with event detection. Figure 1
demonstrates this by presenting a comparison of zero-padded events.
Specifically, the extended MD image obtained, without beamforming, struggled
to capture the first motion because of the similar intensities across upper
and lower spectra, whereas the extended MD image obtained at $\pm$30°angles
performed as expected due to sufficient filtering of the other simultaneous
motion.

Predicted v. Actual |  Class-1 Bend Down (0°) |  Class-2 Sit Down (0°) |  Class-3 Stand Up (0°) |  Class-4 Walk Back (0°) |  Class-5 Walk Back (-30°) |  Class-6 Walk Back (+30°) |  Class-7 Walk Forward (0°) |  Class-8 Walk Forward (+30°) |  Class-9 Walk Forward (-30°)
---|---|---|---|---|---|---|---|---|---
Class-1 Bend Down (0°) |  83.3% |  4.2% |  12.5% |  0% |  0% |  0% |  0% |  0% |  0%
Class-2 Sit Down (0°) |  4.2% |  95.8% |  0% |  0% |  0% |  0% |  0% |  0% |  0%
Class-3 Stand Up (0°) |  0% |  0% |  93.3% |  0% |  0% |  0% |  6.7% |  0% |  0%
Class-4 Walk Back (0°) |  0% |  0% |  0% |  100% |  0% |  0% |  0% |  0% |  0%
Class-5 Walk Back (-30°) |  0% |  0% |  0% |  0% |  100% |  0% |  0% |  0% |  0%
Class-6 Walk Back (+30°) |  0% |  16.7% |  0% |  0% |  0% |  83.3% |  0% |  0% |  0%
Class-7 Walk Forward (0°) |  0% |  0% |  0% |  0% |  0% |  0% |  100% |  0% |  0%
Class-8 Walk Forward (+30°) |  0% |  0% |  0% |  0% |  0% |  0% |  0% |  100% |  0%
Class-9 Walk Forward (-30°) |  0% |  0% |  0% |  0% |  0% |  0% |  0% |  0% |  100%
Table 2: Confusion Matrix for one and two-person data after image segmentation
and zero-padding. The 0°, +30°, and -30° indicate the angle at which the
activity was captured with respect to the radar broadside

##  5 Micro-Doppler Classification

The CNN is a widely used neural network for image classification due to its
ability to automatically select image features through simple convolution and
nonlinear activation operations. The CNN’s architecture, shown in Fig. 1,
utilizes three input modalities including the extended MD images collected
from three angular locations, namely the array broadside and $\pm 30\degree$
with respect to the array broadside. Each extended MD image is of size
$128\times 128\times 3$ and is passed through a 3-layer CNN consisting of 16
filters in each layer, employing $3\times 3$ convolution. It is noted that the
CNN receives the MD image after cropping and zero-padding. As mentioned
previously, individual motions are cropped from extended MD images using
STA/LTA, start/end timestamps. However, the length of the cropped events may
vary depending on the duration of the activity. For instance, walking may have
a longer duration than bending motion. These dimensional mismatches can be
problematic for the CNN since it requires all images to be $128\times
128\times 3$. To address this issue, we zero-pad the cropped images to
$600\times 600\times 3$ and then resize them to $128\times 128\times 3$. For
two-person data, cropping is performed similarly using STA/LTA start/end times
to isolate desired motions.

The outputs of the three networks corresponding to the data inputs are
concatenated and fed to a dense layer followed by an output layer of size $9$,
which represents the number of possible activities. The output layer is a one-
hot encoded vector where the location of a single 1 in the output vector
indicates a specific activity. The internal layers of the network use ReLU
activation function, while the softmax activation function is used for the
output layer.

##  6 Experimental Results

In this section, we show that the radar system is capable of separating the MD
spectrograms of one-person and two-people such that the CNN is able to
adequately classify each individual activity in a transitional movement. We
implemented various algorithms to detect multiple transitional activities
happening within one extended MD image. After pre-processing and spatial
filtering of the extended MD spectrograms, we were able to isolate different
activities happening within the frame at a desired angular region.

To train the CNN, we used 80% of the samples for each class in the training
set, where each sample consists of three $128\times 128\times 3$ images (one
nominal and two beamformed images). The network weights were optimized using
the Adam Optimizer with a learning rate of 0.0001 and 25 epochs. Table 2
presents the confusion matrix, which shows the percentage of each class that
was predicted versus the actual class based on our labeling. It provides a
visual representation of the activities that are being incorrectly classified
as other activities. The overall accuracy of the network is observed to be
around 95.1%.

##  7 Conclusion

In this paper, we introduce a novel approach that utilizes the TF
representation of radar returns obtained from multiple azimuth angles. This
approach enables us to identify combinations of continuous daily activities
that are performed simultaneously at different angles. Our method is effective
in detecting individual activities from extended MD images, isolating them,
and mapping them to their respective angular locations. Notably, our approach
yields promising results, particularly when spectrograms are not entirely
separable based on angles. In conclusion, our approach provides an efficient
means of distinguishing between different motions performed concurrently and
has broad applications in various indoor settings.

##  8 ACKNOWLEDGMENT

The authors would like to thank Jessica Levin, John Barr, Nicholas DePrince,
and Alexander Milazzo for their assistance with data collection.

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# Conditional Bounds on Siegel Zeros

Gautami Bhowmik  G. Bhowmik: Laboratoire Paul Painlevé, Labex-CEMPI,
Université de Lille, 59655 Villeneuve d’Ascq Cedex, France
<EMAIL_ADDRESS>and  Karin Halupczok  K. Halupczok:
Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf,
Universitätsstr. 1, 40225 Düsseldorf, Germany  karin.halupczok@uni-
duesseldorf.de

###### Abstract.

We present an overview of bounds on zeros of $L$-functions and obtain some
improvements under weak conjectures related to the Goldbach problem.

###### Key words and phrases:

Siegel zero, Goldbach problem, congruences, Dirichlet $L$-function,
Generalised Riemann hypothesis

###### 2010 Mathematics Subject Classification:

11P32, 11M26, 11M41

We thank Andrew Granville and Lasse Grimmelt for helpful comments and the
referee for an improved presentation.

Dedicated to Melvyn Nathanson.

##  1\. Introduction

The existence of non-trivial real zeros of a Dirichlet $L$-function would
contradict the Generalised Riemann Hypothesis. One possible counter-example,
called the Landau–Siegel zero, is real and simple and the region in which it
could eventually exist is important to determine. In 1936 Siegel gave a
quantitative estimate on the distance of an exceptional zero from the line
$\Re s=1$. The splitting into cases depending on whether such an exceptional
zero exists or not happens to be an important technique often used in analytic
number theory, for example in the theorem of Linnik. In the first section we
discuss properties of the Siegel zero and results assuming classical and more
recent hypotheses. This part of the paper is expository.

In the second section we present a conditional bound. In 2016 Fei improved
Siegel’s bound for certain moduli under a weakened Hardy–Littlewood conjecture
on the Goldbach problem of representing an even number as the sum of two
primes. In Theorem 11 and Corollary 1 we further weaken this conjecture and
enlarge the set of moduli to include more Dirichlet characters.

##  2\. Background

Consider a completely multiplicative, periodic arithmetic function
$\chi:\mathbb{Z}\rightarrow\mathbb{C}$ where for $q\geq 1$ there exists a
group homomorphism
$\tilde{\chi}:(\mathbb{Z}/q\mathbb{Z})^{\times}\rightarrow\mathbb{C}^{\times}$
such that $\chi(n)=\tilde{\chi}(n(\textrm{mod }q))$ for $n$ coprime to $q$ and
$\chi(n)=0$ if not. We call $\chi$ a Dirichlet character $(\textrm{mod }q)$.
In fact, if $(n,q)=1,\chi(n)$ is a $\phi(q)^{th}$ complex root of unity. We
denote the principal character $\textrm{mod }q$, whose value $\chi(n)$ is
always 1 for $n$ coprime to $q$, by $\chi_{0}\ (\textrm{mod }q)$. The order of
$\chi$ is the least positive integer $n$ such that $\chi^{n}=\chi_{0}$, both
characters having the same modulus. A non-principal character is called
quadratic if $\chi^{2}=\chi_{0}$. In the case where $\chi$ always takes a real
value, the possibilities being only 0 or $\pm 1$, it is called a real
character, otherwise it is called complex. A character modulo $q$ is termed
primitive and $q$ its conductor if it cannot be factored as
${\chi=\chi^{\prime}\chi_{0}},$ where ${\chi_{0}}$ is a principal character
and ${\chi^{\prime}}$ a character of modulus strictly less than $q$. For a
given $\chi\ (\textrm{mod }q)$, there is a unique primitive character
$\tilde{\chi}(\textrm{mod }\tilde{q})$ with least possible $\tilde{q}$, where
$\tilde{q}\mid q$, that induces $\chi$, such that $\chi$ and $\tilde{\chi}$
have the same value at all $n$ coprime to $q$.

The $L$ series were introduced in 1837 by Dirichlet who used them to prove an
analytic formula for the class number and the infinitude of primes in any
arithmetic progression. For $s=\sigma+it$, $\sigma>1$, and a Dirichlet
character $\chi$, we consider the Dirichlet $L$-function

$L(s,\chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=\prod_{p\
\text{prime}}\Big{(}1-\frac{\chi(p)}{p^{s}}\Big{)}^{-1}.$

Note that since $L(s,\chi)=L(s,\tilde{\chi})\prod_{p\mid
q}(1-\tilde{\chi}(p)p^{-s})$ there is no loss in considering only primitive
characters for obtaining analytic properties.

Let $\rho_{\chi}=\beta_{\chi}+i\gamma_{\chi}$ be the non-trivial zeros of the
$L$-function. These are known to be contained in the strip $0<\Re(s)<1$ though
according to the Generalised Riemann Hypothesis (GRH), the only possible value
of $\beta_{\chi}$ is $1/2$. While the GRH remains out of reach, much
unconditional work has been directed towards finding zero-free regions for
$L$-functions.

Around hundred years ago it was proved that real zeros close to $\Re s=1$ are
indeed rare. More precisely,

######  Theorem 1 (Landau–Page).

There is an absolute constant $c>0$ such that for any $Q,T\geq 2$, the product
$\prod_{q\leq Q}\prod_{\chi\mod q}^{\star}L(s,\chi)$ has at most one zero of
an $L$-function in the region

$|t|\leq T,\ 1-\sigma\leq\frac{c}{\log(QT)};$

where ⋆ runs over all primitive real characters of modulus $q$. If such a zero
exists, then it is real and associated to a unique, quadratic $\chi\mod q$.

This eventual ‘bad’ zero contradicting the GRH is called the exceptional or
Siegel or Landau–Siegel zero and the corresponding character is called the
exceptional character. We denote the Landau–Siegel zero by $\beta_{\chi}$ or
simply $\beta$.

The work of Landau and Siegel provide bounds on the proximity of such a zero
on the real axis from $s=1$. Quantitatively,

######  Theorem 2 (Siegel).

For an exceptional zero $\beta$ associated to a primitive character $\chi$ of
conductor $q$ and any $\epsilon>0$ there is a constant $c(\epsilon)>0$ such
that

(2.1) $1-\beta\geq\frac{c(\epsilon)}{q^{\epsilon}}.$

Unfortunately, the constant $c(\epsilon)$ cannot be computed effectively for
any $\epsilon<1/2$, which is a serious difficulty for many applications. In
1951, Tatuzawa [17] did improve on Siegel’s theorem to give an effective
version for almost all cases by proving that for any positive $\epsilon$ there
does exist an effectively computable positive constant $c(\epsilon)$ such that
for all quadratic characters $\chi$, with at most one exception, $L(s,\chi)$
has no zeros in the interval $[1-c(\epsilon)/q^{\epsilon},1].$

###  2.1. Repulsion Property

The possible exceptional zero would force all other zeros, real or otherwise,
of all $L$-functions of the same modulus away from the real axis. We state a
quantitative version of the Deuring–Heilbronn result of 1933-34.

######  Theorem 3.

There exist effective constants $c,c^{\prime}>0$ such that for any $T\geq 2$
and any $q\geq 1$, if for some quadratic $\chi(\textrm{mod }q),L(s,\chi)$ has
an exceptional zero $\beta\in[1-c/\log(qT),1]$, then $\prod_{\chi}L(s,\chi)$,
the product over all characters of modulus $q$ including the exceptional one,
has no other zero in the domain

$\sigma\geq 1-\frac{c^{\prime}|\log((1-\beta)\log qT)|}{\log qT},\ t\leq T.$

Here is a reformulated version of the repulsion phenomenon also due to Linnik
in 1944 which he used to find the size of the least prime in an arithmetic
progression.

######  Theorem 4.

If there exists an exceptional zero $\beta$ with
$1-\beta=\frac{\varepsilon}{\log q}$ for $\varepsilon$ sufficiently small,
then all other zeroes $\sigma+it$ of $L$-functions of modulus $q$ are such
that

$1-\sigma\geq c\frac{\log\frac{1}{\varepsilon}}{\log(q(2+|t|))}$

for an absolute positive constant $c$.

Compared to the classical estimate $\sigma\geq 1-\frac{c}{\log(q(2+|t|))}$
with some absolute positive constant $c$, for a region where $L(s,\chi)$, for
any $\chi\ \textrm{mod }q$, contains no zeros except at most one eventual
exception, we now have a zero-free region wider by a factor of
$\log\frac{1}{\varepsilon}$.

Theorem 4 above was strengthened by Bombieri [4] to

######  Theorem 5.

Let $T\geq 2$ and $\beta$ be an exceptional zero with respect to the otherwise
zero-free region $\sigma\geq\frac{c}{\log T},|t|\leq T$, then there exist
constants $c_{1},c_{2}$ such that if $(1-\beta)\log T\leq c_{2}/e$, then for
any zero $\sigma+it\neq\beta$ of $L(s,\chi)$, we have

$1-\sigma\geq{c_{1}}\frac{\log\frac{c_{2}}{(1-\beta)\log T}}{\log T},$

where $|t|\leq T$ for every primitive $\chi$ of modulus $q\leq T$.

This can be written in terms of a density estimate. Let $N(\alpha,q,T)$ denote
the number of zeroes, counted with multiplicity, of any $L$ function of
modulus $q$ with $\alpha\leq\sigma\leq 1$ and $0\leq t\leq T$ and let
$N^{\prime}$ denote the case when $\beta$ is omitted. Then there is an
improvement

$N^{\prime}(\alpha,q,T)\ll(1-\beta)(\log q)\Big{(}1+\frac{\log T}{\log
q}\Big{)}(qT)^{O(1-\alpha)}$

with effective implied constants over Linnik’s density estimate

$N(\alpha,q,T)\ll(1-\beta)(\log qT)(qT)^{O(1-\alpha)}.$

In [7] Friedlander and Iwaniec state the ‘ultimate Deuring–Heilbronn property’
as

######  Theorem 6.

Let $\chi(\textrm{mod }q)$ be a real primitive character of conductor $q$ with
the largest real zero $\beta$ and let $\eta=\frac{1}{(1-\beta)\log q}\geq 3$.
Then $L(s,\chi)$ has no zeros other than $\beta$ in the region
$\sigma>1-\frac{c\log\eta}{\log q(|t|+1)}$ where $c$ is an absolute positive
constant.

###  2.2. Bounds for $L(1,\chi)$

We know at least since Hecke and Landau that zeros of $L(s,\chi)$ and its
value at $s=1$ are closely related. If $L(1,\chi)$ is sufficiently small
relative to the conductor, then there is a Siegel zero and conversely. More
precisely, if $L(1,\chi)\leq\frac{c}{\log q}$ for a small constant $c>0$, then
$1-\beta\leq\frac{1}{\log q}$. Using the Deuring–Heilbronn repulsion property,
Friedlander and Iwaniec recently proved [7] that

(2.2) $\\{1-\beta\ll(\log q)^{-3}\log\log q\\}\implies\\{L(1,\chi)\ll(\log
q)^{-1}\\}.$

Goldfeld [8] provided an asymptotic result for the location of the Siegel
zero. In fact, when $1-\beta<\frac{c}{\log q}$, he obtained a precise
asymptotic formula

(2.3) $1-\beta\sim\frac{6}{\pi^{2}}L(1,\chi)\Big{(}\sum a^{-1}\Big{)}^{-1}$

where the summation is over all reduced quadratic forms $(a,b,c)$ of
discriminant $-q$.

Dirichlet expressed the value of $L(1,\chi)$ where $\chi(n)=(\frac{-q}{n})$ is
the real primitive character of conductor $q$ in terms of the number $h(q)$ of
equivalence classes of binary quadratic forms of discriminant $q$, which can
equivalently be formulated in terms of the number of ideal classes of an
imaginary quadratic number field $\mathbb{K}=\mathbb{Q}(\sqrt{-}q)$. From the
class number formula $L(1,\chi)=\frac{\pi h(q)}{\sqrt{-q}}$ (here $q<-4$) one
obtains the non-vanishing of $L(1,\chi)$ which is not that obvious when $\chi$
is real, and leads to the prime number theorem in arithmetic progression.
Another obvious consequence of the above formula is the elementary lower bound

$L(1,\chi)\gg\frac{1}{q^{1/2}}.$

Bounding $L(1,\chi)$ is equivalent to estimating the size of the class number
of the imaginary quadratic field $\mathbb{K}=\mathbb{Q}(\sqrt{-}q)$, another
important question in number theory. The corresponding formula is
$L(1,\chi)=\frac{2\pi h_{\mathbb{K}}}{w_{K}\sqrt{d_{\mathbb{K}/\mathbb{Q}}}}$
for the class number $h_{\mathbb{K}}$ and discriminant
$d_{\mathbb{K}/\mathbb{Q}}$ of $\mathbb{K}$, $w_{K}$ the order of the group of
units with regulator being 1 and the LHS being the residue of the Dedekind
zeta function at $s=1$.

Good effective lower bounds are more difficult to obtain. Goldfeld [9] in 1976
using known cases of the Birch and Swinnerton-Dyer conjecture for elliptic
curves showed that

$L(1,\chi)\gg\frac{\log q}{\sqrt{q}(\log\log q)}$

for $q\geq 3$, the implied constant being effective. This together with
Gross–Zagier’s work of 1983 is a major step in the Gauß class number problem.

######  Theorem 7 (Goldfeld–Gross–Zagier).

For every $\epsilon>0$ there exists an effectively computable positive
constant $c$ such that $h(-q)>(c\log q)^{1-\epsilon}$.

This corresponds to a zero-free region of $L(s,\chi)$ of size
$[1-c_{0}\frac{\log^{c_{1}}(q)}{\sqrt{q}},1]$ for some effective positive
constants $c_{0},c_{1}$ and for all real primitive characters.

Oesterlé’s calculation of the involved constant in 1985 makes it possible to
state this bound for $q>0$ as

$L(1,\chi)>\frac{\pi}{55\sqrt{q}}\log q\prod_{p\mid
q}\Big{(}1-\frac{2\sqrt{p}}{p+1}\Big{)}.$

Rather recently Bennett et al. [2] proved that if $\chi$ is a primitive
quadratic character with conductor $q>6677$, then
$L(1,\chi)>\frac{12}{\sqrt{q}}$.

We are still far from the plausible lower bound $L(1,\chi)\gg(\log q)^{-1}$
which holds in many cases, for example for complex characters with an
effective constant.

Aisleitner et al. [1] in 2019 showed the existence, for $q$ sufficiently
large, of an extremal non-principal character which satisfies, for constants
$C$ and $\gamma$, $|L(1,\chi)|\geq e^{\gamma}(\log\log q+\log\log\log q-C)$
using the method of resonance for detecting large values of the Riemann zeta
function. Up to the constant, this corresponds to the predicted order of the
extremal values.

A simple unconditional upper bound is $L(1,\chi)\ll\log q$. The implied
constants have been worked on by a variety of methods. For example, for
complex characters Granville and Soundararajan [10] determine the constant
$c_{k}$ for primitive characters of order $k$ for which the bounds
$|L(1,\chi)|\leq(c_{k}+o(1))\log q$ hold true. For real primitive characters,
the constant $c_{2}=\frac{1}{4}(2-\frac{2}{\sqrt{e}}+o(1))\log q$ was obtained
by Stephens for prime characters [16] and Pintz [14] extended this to non-
prime characters.

####  2.2.1. Conditional Bounds

The optimal bounds of $L(1,\chi)$ under the condition of GRH are

$(\log\log q)^{-1}\ll L(1,\chi)\ll\log\log q$

where the implied constants are effective. Precisely speaking,

######  Theorem 8 (Littlewood 1928).

If the Generalised Riemann Hypothesis is true then

$\Big{(}\frac{1}{2}+o(1)\Big{)}\frac{\pi^{2}}{6e^{\gamma}\log\log q}\leq
L(1,\chi)\leq(2+o(1))e^{\gamma}\log\log q$

where $\gamma$ is Euler’s constant.

Only the implied constants in the above can be improved because there actually
exist infinitely many $q$ for which the special value of the corresponding
character at $s=1$ correspond to the above magnitude of orders. The classical
unconditional $\Omega$ results, that
$(1+o(1))\frac{\pi^{2}}{6e^{\gamma}\log\log q}\geq L(1,\chi)$ and
$L(1,\chi)\geq(1+o(1))e^{\gamma}\log\log q$ hold for infinitely many $q$ [5]
show that there is a factor of 2 that remains undetermined for the extreme
values.

We cite one example of a recent refined upper and lower bound established by
Lamzouri et al. [12] assuming the GRH for characters of large conductor and
studying certain character sums. For $q\geq 10^{10}$, the bounds obtained
therein can be written in a simplified manner as

$\frac{\pi^{2}}{12e^{\gamma}\log\log q}<|L(1,\chi)|<2e^{\gamma}\log\log q.$

The lower bound can be improved a lot by admitting the existence of
Landau–Siegel zeros, and thus weakening the GRH. One such assumption,
sometimes called the Modified Generalised Riemann Hypothesis (MGRH), is that
all the zeros of $L(s,\chi)$ lie either on the critical line or on the real
axis. Sarnak and Zaharescu [15] showed that if all Dirichlet $L(s,\chi)$ with
$\chi$ real satisfy the MGRH then

$L(1,\chi)\geq\frac{c^{\epsilon}}{(\log|q|)^{\epsilon}}$

for any positive $\epsilon$. The above constant is ineffective but the bounds
can be made effective under certain additional conditions. These bounds use
the explicit formula with an appropriately constructed kernel function.

Assuming that the GRH holds except for one possible exception, Friedlander and
Iwaniec obtained an improved version of (2.2). They proved that:

######  Theorem 9 ([7]).

Let the GRH be true except for only one $\beta>3/4$. Now if
$1-\beta\ll(\log\log q)^{-1}$, then

$1-\beta\ll L(1,\chi)\ll(1-\beta)(\log\log q)^{2}.$

##  3\. Better Siegel zero bounds from Weak Goldbach conjectures

Connections between Siegel zeros and the Goldbach problem were studied, for
example in [3] and [6]. Among the classical conjectures of the Goldbach
problem is one due to Hardy and Littlewood in 1923 that predicts an
equivalence between the number of representations of an even number as a sum
of two primes and a singular series, $g(n)=\leavevmode\nobreak\
\sum_{n=p_{1}+p_{2}}1\sim\mathcal{S}(n)$, where

$\mathcal{S}(n):=\frac{n}{\varphi(n)}\prod_{p\nmid
n}\Big{(}1-\frac{1}{(p-1)^{2}}\Big{)}\cdot\frac{n}{\log^{2}n}=2C_{2}\Big{(}\prod_{\begin{subarray}{c}p\mid
n\\\ p>2\end{subarray}}\frac{p-1}{p-2}\Big{)}\cdot\frac{n}{\log^{2}n}$

with the twin prime constant

$C_{2}=\prod_{p>2}\Big{(}1-\frac{1}{(p-1)^{2}}\Big{)}$

which is approximately 0.66.

Fei [6] obtained an upper bound for $\beta$ under a weakened form of the
Hardy–Littlewood conjecture (WHL), namely

######  Conjecture 1 (WHL).

There exists a positive contant $\delta$ such that $g(n)\geq\frac{\delta
n}{\log^{2}n}$ for every even integer $n>2$.

Considering the size of $\mathcal{S}(n)$ we could even expect a $\delta>1.32$,
but in this weakened form of the Hardy–Littlewood conjecture, we assume only
the existence of some small positive $\delta$.

We now state Fei’s theorem.

######  Theorem 10 ([6]).

If the WHL-conjecture is true and if there is an exceptional zero $\beta$ for
a character $\chi$ with a prime modulus $q\equiv 3\mod 4$, then there exists a
positive constant $c$ such that $1-\beta\geq\frac{c}{\log^{2}q}$.

Here, the corresponding region for the exceptional zero $\beta$ is meant to be
that of Theorem 1 with $T=q$. We will keep to this convention for the rest of
this section.

One would like to know if it is possible to include other moduli in the above
result or to relax the assumed WHL-conjecture.

Here we generalise Fei’s result in these two aspects and obtain a conditional
improvement of Siegel’s bound (Theorem 2) for certain exceptional characters
which includes Fei’s modulus condition (Corollary 1). Our result still assumes
the weak Hardy–Littlewood conjecture but allows certain exceptions (WHLE)
making it weaker than the WHL. Our proof is similar to that of Fei’s but
exploits, apart from the use of the WHLE, the generalisation to suitable
composite moduli $q$.

######  Conjecture 2 (WHLE).

Suppose that $x$ is sufficiently large, and $q\leq x/4$. Then we have, with at
most $x/8q$ exceptions,

$g(n)\gg\frac{n}{\log^{2}n}$

for the multiples $n$ of $q$ in the interval $x/2<n\leq x$.

######  Theorem 11.

Assume the WHLE Conjecture to be true. Let $q$ be a sufficiently large integer
and $\chi$ be a primitive character mod $q$ with $\chi(-1)=-1$ such that there
is an exceptional zero $\beta$ of $L(s,\chi)$. Then there exists an effective
constant $c>0$ such that $1-\beta\geq\frac{c\varphi(q)}{q\log^{2}(q)}$.

###### Proof of Theorem 11.

Step 1. We prove the following lower bound for the sum

(3.1) $S=\sum_{k=1}^{q}\Big{(}\sum_{2<p\leq
x}\mathrm{e}\Big{(}\frac{kp}{q}\Big{)}\Big{)}^{2}\geq\frac{\delta
x^{2}}{8\log^{2}x}$

for any sufficiently large real $x>2$ and some small $\delta>0$.

For this, we first note note that

$\displaystyle S$ $\displaystyle=\sum_{k=1}^{q}\sum_{2<p_{1},p_{2}\leq
x}\mathrm{e}\Big{(}\frac{k(p_{1}+p_{2})}{q}\Big{)}$ $\displaystyle=\sum_{n\leq
2x}\sum_{k=1}^{q}\mathrm{e}\Big{(}\frac{kn}{q}\Big{)}\sum_{\begin{subarray}{c}2<p_{1},p_{2}\leq
x\\\ p_{1}+p_{2}=n\end{subarray}}1=\sum_{\begin{subarray}{c}n\leq 2x\\\
n\equiv 0(q)\end{subarray}}q\sum_{\begin{subarray}{c}2<p_{1},p_{2}\leq x\\\
p_{1}+p_{2}=n\end{subarray}}1.$

Let $x$ be large enough with

(3.2) $q\leq x/4.$

Hence under the assumption of Conjecture 2, for all even $n$ in the interval
$x/2<n\leq x$ that are divisible by $q$, we have

$\sum_{\begin{subarray}{c}2<p_{1},p_{2}\leq x\\\
p_{1}+p_{2}=n\end{subarray}}1\geq\delta\frac{x}{\log^{2}x}$

for some constant $\delta>0$, with the possible exception of at most $x/8q$
such $n$. Let $\mathcal{E}$ be the set of these exceptions.

Keeping this in mind, we obtain the lower bound

(3.3) $S\geq q\sum_{\begin{subarray}{c}x/2<n\leq x\\\ 2|n\\\ n\equiv
0(q)\end{subarray}}\ \sum_{p_{1}+p_{2}=n}1\geq
q\sum_{\begin{subarray}{c}x/2<n\leq x\\\ 2|n\\\ n\equiv 0(q)\\\
n\notin\mathcal{E}\end{subarray}}\delta\frac{x}{\log^{2}x}\\\
\geq\frac{q\delta
x}{\log^{2}x}\Big{(}\frac{x}{4q}-\frac{x}{8q}\Big{)}=\frac{\delta
x^{2}}{8\log^{2}x}$

as was to be shown.

Step 2. We now prove a more explicit expression for the sum $S$ in the first
step.

In (3.1), the sum over $p$ is subdivided into parts depending on whether or
not $q$ is divisible by $p$, i.e.

$S_{1}=\sum_{2<p\leq
x}\mathrm{e}\Big{(}\frac{kp}{q}\Big{)}=\sum_{\begin{subarray}{c}2<p\leq x\\\
p\nmid
q\end{subarray}}\mathrm{e}\Big{(}\frac{kp}{q}\Big{)}+\sum_{\begin{subarray}{c}2<p\leq
x\\\ p\mid
q\end{subarray}}\mathrm{e}\Big{(}\frac{kp}{q}\Big{)}=\sum_{\begin{subarray}{c}2<p\leq
x\\\
(p,q)=1\end{subarray}}\mathrm{e}\Big{(}\frac{kp}{q}\Big{)}+\mathcal{O}(\omega(q))\\\
=\sum_{\begin{subarray}{c}1\leq a\leq q\\\
(a,q)=1\end{subarray}}\mathrm{e}\Big{(}\frac{ka}{q}\Big{)}\sum_{\begin{subarray}{c}2<p\leq
x\\\ p\equiv
a(q)\end{subarray}}1+\mathcal{O}(\omega(q))=c_{q}(k)\sum_{\begin{subarray}{c}2<p\leq
x\\\ p\equiv a(q)\end{subarray}}1+\mathcal{O}(\omega(q)),$

where $c_{q}(k)$ denotes the above Ramanujan sum

$\sum_{\begin{subarray}{c}1\leq a\leq q\\\
(a,q)=1\end{subarray}}e({\tfrac{ka}{q}})$

while $\omega(q)=\\#\\{p\mid q\\}$. Using the prime number theorem in
arithmetic progressions [13], the last sum over $p$ is written out as

(3.4) $\sum_{\begin{subarray}{c}2<p\leq x\\\ p\equiv
a(q)\end{subarray}}1=\frac{\operatorname{li}(x)}{\varphi(q)}-\frac{\chi(a)}{\varphi(q)}\int_{2}^{x}\frac{u^{\beta-1}}{\log
u}du+\mathcal{O}(x\exp(-\tilde{c}\sqrt{\log x}))$

for some constant $\tilde{c}>0$, which holds uniformly in

(3.5) $q\leq\exp(C\sqrt{\log x})$

for any $C>0$ (this range is consistent with (3.2), though we could have
chosen a larger $x$). Hence, this gives

(3.6)
$S_{1}=M+c_{q}(k)\frac{\operatorname{li}(x)}{\varphi(q)}+\mathcal{O}\Big{(}qx\exp(-\tilde{c}\sqrt{\log
x})\Big{)}$

with term $M$ being

$M=\frac{-1}{\varphi(q)}\sum_{a=1}^{q}\mathrm{e}\Big{(}\frac{ak}{q}\Big{)}\chi(a)\int_{2}^{x}\frac{u^{\beta-1}}{\log
u}du=\frac{-\tau_{k}(\chi)}{\varphi(q)}\int_{2}^{x}\frac{u^{\beta-1}}{\log
u}du,$

since $\chi(a)=0$ if $(a,q)>1$, with the Gauß sum

$\tau_{k}(\chi)=\sum_{a=1}^{q}\mathrm{e}\Big{(}\frac{ak}{q}\Big{)}\chi(a).$

Inserting the expansion $\int_{2}^{x}\frac{u^{\beta-1}}{\log
u}du=\frac{x^{\beta}}{\beta\log x}+\mathcal{O}(\frac{x^{\beta}}{\log^{2}x})$
yields

$M=\frac{-\tau_{k}(\chi)}{\varphi(q)}\cdot\frac{x^{\beta}}{\beta\log
x}+\mathcal{O}\Big{(}\frac{q^{1/2}}{\varphi(q)}\cdot\frac{x^{\beta}}{\log^{2}x}\Big{)}$

from the estimate $\tau_{k}(\chi)\ll q^{1/2}$.

We substitute this expression in (3.6) and the resulting approximation for
$S_{1}$ into $S$ to get

(3.7)
$S=\sum_{k=1}^{q}S_{1}^{2}=\sum_{k=1}^{q}\Big{(}c_{q}(k)\frac{\operatorname{li}(x)}{\varphi(q)}-\frac{\tau_{k}(\chi)}{\varphi(q)}\cdot\frac{x^{\beta}}{\beta\log
x}\\\ \hfill+\mathcal{O}\Big{(}qx\exp(-\tilde{c}\sqrt{\log
x})+\frac{q^{1/2}}{\varphi(q)}\cdot\frac{x^{\beta}}{\log^{2}x}\Big{)}\Big{)}^{2}\\\
=\sum_{k=1}^{q}\Big{(}c^{2}_{q}(k)\frac{\operatorname{li}^{2}(x)}{\varphi^{2}(q)}+\frac{\tau^{2}_{k}(\chi)}{\varphi^{2}(q)}\cdot\frac{x^{2\beta}}{\beta^{2}\log^{2}x}\Big{)}\\\
+\mathcal{O}\Big{(}\frac{q^{1/2}x^{\beta}}{\varphi(q)\log
x}q^{2}x\exp(-\tilde{c}\sqrt{\log
x})+\frac{q^{2}x^{2\beta}}{\beta\varphi^{2}(q)\log^{3}x}+q^{3}x^{2}\exp(-2\tilde{c}\sqrt{\log
x})\Big{)},$

where all the mixed terms containing the Ramanujan sum have now disappeared
since

$\sum_{k=1}^{q}c_{q}(k)\tau_{k}(\chi)=\sum_{k=1}^{q}\sum_{\begin{subarray}{c}1\leq
a\leq q\\\ (a,q)=1\end{subarray}}\sum_{\begin{subarray}{c}1\leq b\leq q\\\
(b,q)=1\end{subarray}}\mathrm{e}\Big{(}\frac{ak}{q}\Big{)}\mathrm{e}\Big{(}\frac{bk}{q}\Big{)}\chi(b)\\\
=\sum_{\begin{subarray}{c}1\leq a\leq q\\\
(a,q)=1\end{subarray}}\sum_{\begin{subarray}{c}1\leq b\leq q\\\ (b,q)=1\\\
b\equiv-a(q)\end{subarray}}\chi(b)q=q\sum_{\begin{subarray}{c}1\leq a\leq q\\\
(a,q)=1\end{subarray}}\chi(-a)=0$

and $\sum_{k=1}^{q}c_{q}(k)=0$. The $\mathcal{O}$-term in (3.7) simplifies to

(3.8)
$E_{expl}=\mathcal{O}\Big{(}\frac{x^{2}}{\log^{3}x}\frac{q^{2}}{\varphi(q)^{2}}+q^{3}x^{2}\exp(-\tilde{c}\sqrt{\log
x})\Big{)}.$

For the main term in (3.7), we use properties of Gauß sums ([13, p.287]),

$\tau_{k}(\chi)=\Big{\\{}\begin{aligned} \bar{\chi}(k)\tau_{1}(\chi),&\
(k,q)=1\\\ 0,&\ \text{ else},\\\ \end{aligned}$

so that the sum over $\tau_{k}^{2}(\chi)$ in (3.7) becomes

$\frac{1}{\varphi^{2}(q)}\sum_{\begin{subarray}{c}k=1\\\
(k,q)=1\end{subarray}}^{q-1}\tau_{1}^{2}(\chi)\bar{\chi}^{2}(k)=\frac{q}{\varphi(q)}\chi(-1)$

since $\tau^{2}_{1}(\chi)=\chi(-1)q$ and $\chi^{2}=\chi_{0}$. Similarly, we
have

$\sum_{k=1}^{q}c_{q}^{2}(k)=q\varphi(q),$

see [13, p.113]. Hence

(3.9)
$S=\frac{q}{\varphi(q)}\operatorname{li}^{2}(x)+\frac{q}{\varphi(q)}\chi(-1)\frac{x^{2\beta}}{\beta^{2}\log^{2}x}+E_{expl}.$

######  Comment.

We note that an alternative approach via the identity

$\frac{q}{\varphi(q)}\sum_{\chi(q)}\chi(-1)|\psi(x,\chi)|^{2}=\sum_{k=1}^{q}\Big{(}\sum_{p\leq
x}\Lambda(p)\mathrm{e}\Big{(}\frac{kp}{q}\Big{)}\Big{)}^{2}+o(\log^{3}x)$

and the use of an explicit formula for $\psi(x,\chi)$ might avoid the use of
Gauß sums in Step 2 of the proof.

Step 3. Now we compare the lower bound from Step 1 with the explicit
evaluation from Step 2. With the assumption $\chi(-1)=-1$ we get the
inequality

$\frac{x^{2\beta}}{\beta^{2}\log^{2}x}\leq\Big{(}1-\frac{\delta}{8}\cdot\frac{\varphi(q)}{q}\Big{)}\frac{x^{2}}{\log^{2}x}+E_{\mathrm{expl}},$

where $E_{\mathrm{expl}}$ is the error term of (3.9) above. This yields

$x^{2\beta-2}\leq\Big{(}1-\frac{\delta}{8}\cdot\frac{\varphi(q)}{q}\Big{)}+\mathcal{O}\Big{(}\frac{q}{\varphi(q)\log
x}+q^{3}\exp(-\tilde{c}\sqrt{\log x})\Big{)}.$

We may now choose $x$ such that $(\frac{4\log q}{\tilde{c}})^{2}\leq\log x\leq
c_{3}\log^{2}q$ for some $c_{3}>(4/\tilde{c})^{2}$, so that $x$ is not too
large compared to $q$, but still such that the choice is admissible with our
previous assumptions (3.2) and (3.5). Hence with $1/\log
x\leq\tilde{c}^{2}/16\log^{2}q$, we obtain

$x^{2\beta-2}\leq\Big{(}1-\frac{\delta}{8}\cdot\frac{\varphi(q)}{q}\Big{)}+\frac{c_{1}}{\log^{2}q}\frac{q}{\varphi(q)}$

for some positive constant $c_{1}$, since the expression $q/\varphi(q)\log x$
dominates the error term due to

$q^{3}\exp(-\tilde{c}\sqrt{\log x})\leq
q^{3}\exp\Big{(}-\tilde{c}\frac{4}{\tilde{c}}\log q\Big{)}=q^{3}\exp(-4\log
q)=q^{-1}.$

Assume now that $q$ is large enough so that
$\frac{16c_{1}}{\log^{2}q}\leq\delta\frac{\varphi^{2}(q)}{q^{2}}$. This means
that

$x^{2\beta-2}\leq\Big{(}1-\frac{\delta}{8}\cdot\frac{\varphi(q)}{q}\Big{)}+\frac{\delta}{16}\frac{\varphi(q)}{q}\leq
1-\frac{\delta}{16}\frac{\varphi(q)}{q}$

for $\delta<8$. This gives the inequality of Theorem 11, since

$\beta-1\leq\frac{\log(1-\frac{\delta}{16}\cdot\frac{\varphi(q)}{q})}{2\log
x}\leq\frac{-c\varphi(q)}{q\log^{2}q}\text{ with }c=\frac{\delta}{32c_{3}}>0,$

where we use the upper bound for $\log x$, which is $\log x\leq
c_{3}\log^{2}q$. ∎

We emphasize that all constants in the above proof are effectively computable
since the constant $\tilde{c}$ in (3.4) coming from the prime number theorem
in progressions is itself so and all other constants in the proof can be
chosen effectively depending on $\tilde{c}$.

The next corollary gives a criterion for a composite modulus $q$ to satisfy
Theorem 11.

######  Corollary 1.

Assume the WHLE Conjecture and that $q$ is a sufficiently large integer with
$\\#\\{t\mid q\\}\cap(\\{4\\}\cup\\{p\equiv 3(\textrm{mod }4);\ p\text{
prime}\\})=1$. If there is an exceptional zero $\beta$ for a character mod
$q$, then $1-\beta\geq\frac{c\varphi(q)}{q\log^{2}q}$ for some (effective)
constant $c>0$.

###### Proof.

For the moduli $q$ in question there is a single real primitive character such
that $\chi(-1)=-1$. This is certainly true if
$q\in\mathcal{S}:=\\{4\\}\cup\\{p\equiv 3(\textrm{mod }4)\\}$, and $q=tm$ with
$t\in\mathcal{S}\cup\\{2\\}$ and with $m$ having only prime divisors $p\equiv
1(\textrm{mod }4)$. Then there is only a single real $\chi$ with
$\chi(-1)=-1$, namely the one that is induced by that mod $t$. For the others,
$\chi(-1)=1$, since this equation holds for every prime modulus $p\equiv
1(\textrm{mod }4)$.

Hence, if we assume that $\beta$ exists for an exceptional character $\chi$,
we know that $\chi$ is real and primitive, and necessarily $\chi(-1)=-1$. Then
Theorem 11 applies to give the assertion. ∎

We recover Fei’s Theorem from Corollary 1 when $q=p\equiv 3(\textrm{mod }4)$.

Under GRH except for a possible $\beta>3/4$ and the WHLE Conjecture we can
deduce the conditional bound

$L(1,\chi)\gg 1-\beta\gg\frac{\varphi(q)}{q\log^{2}q}.$

using Theorem 9 of Friedlander and Iwaniec, and using Theorem 11, supposing
$\chi(-1)=-1$ for the exceptional character $\chi$ mod $q$. In fact, with
Goldfeld’s asymptotic formula (2.3) we can relax the assumption of GRH with
one exception to obtain

######  Corollary 2.

Let the WHLE Conjecture be true. Assuming $L(1,\chi)=o(\log^{-1}q)$, we have
$L(1,\chi)\gg R_{q}\varphi(q)q^{-1}\log^{-2}q$ for an exceptional character
$\chi$ mod $q$ with $\chi(-1)=-1$. Here $R_{q}=\sum_{(a,b,c)}a^{-1}$ with the
sum going over all reduced quadratic forms $(a,b,c)$ of discriminant $-q$.

A reduced quadratic form of discriminant $-q$ is an integer triple $(a,b,c)$
with $b^{2}-4ac=-q$ and $-a<b\leq a<\frac{1}{4}\sqrt{q}$, see [9, p. 624].

###### Proof.

By (2.3) from [9, p. 624] and Theorem 11, we have

$L(1,\chi)\sim\frac{\pi^{2}}{6}\Big{(}\sum_{(a,b,c)}a^{-1}\Big{)}(1-\beta)\gg\frac{R_{q}\varphi(q)}{q\log^{2}q}.$

∎

Note that for a prime modulus $q=p\equiv 3(\textrm{mod }4)$, we have
$R_{q}=\sum_{(a,b,c)}a^{-1}\geq 1$ since then there is a reduced quadratic
form $(1,1,c)$ with $a=1$. Then our bound states $L(1,\chi)\gg\log^{-2}q$
under the assumptions of Corollary 2.

###  3.1. Questions

We would like to know if the case $\chi(-1)=1$ could be handled as well. But
we have not been able to combine the results of [3] together with Fei’s
approach which seems to be a natural way to proceed in this direction.

One could also ask if the WHLE Conjecture itself can be obtained from existing
results. We were not able to find anything appropriate. Even when averaging
the assertion over moduli $q\leq Q$, we only reach a special case of
Conjecture 1 from [11], which seems to be out of reach.

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