---
abstract: 'In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in Omar and Mazhouda, J. Number Theory 125 (2007) no.1, 50-58; J. Number Theory 130 (2010) no.4, 1109-1114. Furthermore, we formulate a criterion for the partial Riemann hypothesis and we provide some numerical evidence for it using new formulas for the Li coefficients.'
address:
- 'Faculty of Science of Tunis, Department of Mathematics, 2092 Tunis, Tunisia'
- 'Faculty of Science of Tunis, Department of Mathematics, 2092 Tunis, Tunisia'
- 'Faculty of Science of Monastir, Department of Mathematics, 5000 Monastir, Tunisia'
author:
- 'Sami Omar, Raouf Ouni, Kamel Mazhouda'
title: 'On the zeros of Dirichlet $L$-functions'
---

[^1]

[Introduction]{}\[sec.1\] The Li criterion for the Riemann hypothesis (see. [@6]) is a necessary and sufficient condition that the sequence $$\lambda_{n}=\sum_{\rho}\left[1-\left(1-\frac{1}{\rho}\right)^{n}\right]$$ is non-negative for all $n\in{\mathbb {N}}$ and where $\rho$ runs over the non-trivial zeros of $\zeta(s)$. This criterion holds also for the Dirichlet $L$-functions and for a large class of Dirichlet series, the so called the Selberg class as given in [@10]. More recently, Omar and Bouanani [@9] extended the Li criterion for function fields and established an explicit and asymptotic formula for the Li coefficients.\
Numerical computation of the first 100 of the Li coefficients $\lambda_{n}$ which appear in this criterion was made by Maslanka [@8] and later by Coffey in [@2], who computed and verified the positivity of about 3300 of the Li coefficient $\lambda_{n}$. The main empirical observation made by Maslanka is that these coefficients can be separated in two parts, where one of them grows smoothly while the other one is very small and oscillatory. This apparent smallness is quite unexpected. If it persisted until infinity then the Riemann hypothesis would be true. As we said above, this criterion was extended to a large class of Dirichlet series [@10] and no calculation or verification of the positivity to date in the literature made for other $L$-functions.\
In this paper, we compute and verify the positivity of the Li coefficients for the Dirichlet $L$-functions using an arithmetic formula established in [@10; @11]. Furthermore, we formulate a criterion for the partial Riemann hypothesis. Additional results are presented, including new formulas for the Li coefficients. Basing on the numerical computations made below, we conjecture that these coefficients are increasing in $n$. Should this conjecture hold, the validity of the Riemann hypothesis would follow.\
Next, we review the Li criterion for the case of the Dirichlet $L$-functions. Let $\chi$ be a primitive Dirichlet character of conductor $q$. The Dirichlet $L$-function attached to this character is defined by $$L(s, \chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}, \ \ \ \ (Re(s) > 1).$$ For the trivial character $\chi = 1$, $L(s, \chi)$ is the Riemann zeta function. It is well known [@3] that if $\chi\neq1$ then $L(s,\chi)$ can be extended to an entire function in the whole complex plane and satisfies the functional equation $$\xi(s,\chi)= \omega_{\chi}\xi(1-s,\overline{\chi}),$$ where $$\xi(s,\chi)=\left(\frac{q}{\pi}\right)^{(s+a)/2}\Gamma\left(\frac{s+a}{2}\right)L(s,\chi),$$ $$a=\left\{\begin{array}{crll}0&\hbox{if}&\chi(-1)= 1\\1&\hbox{if}&\chi(-1) =-1,\end{array}\right.\ \ \ \ \hbox{and}\ \ \ \  \omega_{\chi}=\frac{\tau(\chi)}{\sqrt{q}i^{a}},$$ where $\tau(\chi)$ is the Gauss sum $$\tau(\chi)=\sum_{m=1}^{q}\chi(m)e^{2\pi im/q}.$$ The function $\xi(s,\chi)$ is an entire function of order one. The function $\xi(s,\chi)$ has a product representation $$\label{eq.1}\xi(s,\chi) =\xi(0,\chi)\prod_{\rho}\left(1-\frac{s}{\rho}\right),$$ where the product is over all the zeros of $\xi(s,\chi)$ in the order given by $|Im(\rho)|<T$ for $T\rightarrow\infty$. If $N_{\chi} (T)$ counts the number of zeros of $L(s,\chi)$ in the rectangle $0\leq Re(s)\leq1$, $0<Im(s)\leq T$ (according to multiplicities) one can show by standard contour integration the formula $$N_{\chi}(T)=\frac{1}{2\pi}T\log T+c_{1}T+O\left(\log T\right),$$ where $$c_{1}=\frac{1}{2\pi}\left(\log q-\left(\log(2\pi)+1\right)\right).$$ We put $$\lambda_{\chi}(n)=\sum_{\rho}\left[1-\left(1-\frac{1}{\rho}\right)^{n}\right],$$ where the sum over $\rho$ is $\sum_{\rho}=\lim_{T\mapsto\infty}\sum_{|Im\rho|\leq T}$.\
[**Li’s criterion**]{} says that $\lambda_{\chi}(n)> 0$ for all $n = 1, 2, .$ . . if and only if all of the zeros of $\xi(s,\chi)$ are located on the critical line $Re(s) = 1/2$.\
The paper is organized as follows. In Section \[sec.2\], we recall the arithmetic formula for the Li coefficients for the Dirichlet $L$-functions and we give an estimate for the error term of $\lambda_{\chi}(n)$. In Section \[sec.3\], we show that $\lambda_{\chi}(n)\geq0$ if every non-trivial zero of $L(s,\chi)$ with $|Im(\rho)|<\sqrt{n}$ satisfies $Re(\rho)=1/2$ (that is partial Riemann hypothesis) and we give an estimate for the difference $|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)|$, where $\lambda_{\chi}(n,T)$ are the partial Li coefficients. In Section \[sec.4\], we prove new formulas (integral and summation formula) for the Li coefficients $\lambda_{\chi}(n)$. Finally, in Section \[sec.5\], we report numerical computations of the Li coefficients using different formulas established in the previous sections unconditionally or under the Riemann hypothesis.

Li’s coefficients {#sec.2}
=================

Applying [@10 Theorem 2.2] for the case of the Dirichlet $L$-functions, we get the following arithmetic formula.

\[th.1\]Let $\chi$ be a primitive Dirichlet character of conductor $q>1$. We have $$\begin{aligned}
\label{eq.2}
\lambda_{\chi}(n)&=&-\sum_{j=1}^{n}(_{j}^{n})\frac{(-1)^{j-1}}{(j-1)!}\sum_{k=1}^{+\infty}\frac{\Lambda(k)}{k}\chi(k)(\log k)^{j-1}\nonumber\\
&&\ \ +\ \ \frac{n}{2}\left(\log\frac{q}{\pi}-\gamma\right)+\tau_{\chi}(n)\end{aligned}$$ where $$\tau_{\chi}(n)=\left\{\begin{array}{crll}\sum_{j=2}^{n}(_{j}^{n})(-1)^{j}\left(1-\frac{1}{2^{j}}\right)\zeta(j)-\frac{n}{2}\sum_{l=1}^{+\infty}\frac{1}{l(2l-1)}&\hbox{if}&\chi(-1)=1,\\ \sum_{j=2}^{n}(_{j}^{n})(-1)^{j}2^{-j}\zeta(j)&\hbox{if}&\chi(-1)=-1.\end{array}\right.$$

Theorem \[th.1\] is also proved by Coffey in [@2] and Li in [@7]. The arithmetic formula above can be written as $$\begin{aligned}
\label{eq.3}
\lambda_{\chi}(n)&=&\left[\log\frac{q}{\pi}+\psi\left(\frac{a+1}{2}\right)\right]\frac{n}{2}+\sum_{j=2}^{n}(_{j}^{n})\frac{1}{(j-1)!}2^{-j}\psi^{(j-1)}\left(\frac{a+1}{2}\right)\nonumber\\
&&-\ \sum_{j=1}^{n}(_{j}^{n})\frac{(-1)^{j-1}}{(j-1)!}\sum_{k=1}^{+\infty}\frac{\Lambda(k)\chi(k)}{k}(\log k)^{j-1},\nonumber\end{aligned}$$ where $a=0$ if $\chi(-1)=1$ and 1 if $\chi(-1)=-1$, with $\psi(\frac{1}{2})=-\gamma-2\log2$, $\psi(1)=-\gamma$,  $\gamma$ is the Euler constant, $\psi^{(j-1)}(1)=(-1)^{j}(j-1)!\zeta(j)$   and $\psi^{(j-1)}(\frac{1}{2})=(-1)^{j+1}j!(2^{j+1}-1)\zeta(j+1).$ Here, $\psi=\frac{\Gamma'}{\Gamma}$ denotes the digamma function.\
An asymptotic formula for the number $\lambda_{\chi}(n)$ was proved in [@12 Theorem 3.1] using the arithmetic formula. Furthermore, it is equivalent to the Riemann hypothesis.

\[th.2\] We have $$RH\Leftrightarrow\lambda_{\chi}(n)=\frac{1}{2}n\log n+c_{\chi}n+O(\sqrt{n}\log n),$$ where $$c_{\chi}=\frac{1}{2}(\gamma-1)+\frac{1}{2}\log(q/\pi),$$ and $\gamma$ is the Euler constant.

Here, we estimate the error term for $\lambda_{\chi}(n)$ using the arithmetic formula (\[eq.2\]). Writing (\[eq.2\]) in the form $$\lambda_{\chi}(n)=\tilde{\lambda}_{\chi}(n,M)+E_{M},$$ where $$\label{eq.4}\tilde{\lambda}_{\chi}(n,M)=\frac{n}{2}\left(\log\frac{q}{\pi}-\gamma\right)+\tau_{\chi}(n)-\sum_{j=1}^{n}(_{j}^{n})\frac{(-1)^{j-1}}{(j-1)!}\sum_{k\leq M}\frac{\Lambda(k)}{k}\chi(k)(\log k)^{j-1}$$ and $$E_{M}=-\sum_{j=1}^{n}(_{j}^{n})\frac{(-1)^{j-1}}{(j-1)!}\sum_{k>M}\frac{\Lambda(k)}{k}\chi(k)(\log k)^{j-1}=-\sum_{k>M}\frac{\Lambda(k)}{k}\chi(k)L_{n-1}^{1}(\log k),$$ where $L_{n-1}^{1}$ is an associated Laguerre polynomial of degree $n-1$.\
Next, for our computation in the section \[sec.5\], we need to find $M$ such that $|E_{M}|\leq 10^{-\nu}$. An estimate for the Laguerre polynomials is due to Koepf and Schmersau [@5 Theorem 2]. Actually, they have shown that $$|L_{n}^{\alpha}(x)|<e^{x/2}\left[(n+\alpha)/x\right]^{\alpha/2}$$ for $x\in{[0,4(n+\alpha)]}$ when $n+\alpha>0$ and $\alpha$ is an integer. Then, we obtain $$\begin{aligned}
\label{eq.33}|E_{M}|&\leq&\left|\sqrt{\frac{n}{\log M}}\sum_{m>M}^{+\infty}\frac{\Lambda(m)}{\sqrt{m}}\chi(m)\right|\nonumber\\
 &\leq&\sqrt{\frac{n}{\log M}}\left\{\left|\sum_{p>M}^{+\infty}\frac{\log p}{\sqrt{p}}\chi(p)\right|+\left|\sum_{p^{2}>M}^{+\infty}\frac{\log p}{p}\chi(p^{2})\right|\right\}\nonumber\\
 &&+\ \ \sqrt{\frac{n}{\log M}}\left|\sum_{p^{j}>M,j>2}^{+\infty}\frac{\log p}{p^{j/2}}\chi(p^{j})\right|.
 \end{aligned}$$ We have $|\chi(p^{j})|<1$. Therefore, $$\left|\sum_{p>M}^{+\infty}\frac{\log p}{\sqrt{p}}\chi(p)\right|+\left|\sum_{p^{2}>M}^{+\infty}\frac{\log p}{p}\chi(p^{2})\right|\leq\left|\sum_{p>M}^{+\infty}\frac{\log p}{\sqrt{p}}\chi(p)\right|+\left|\sum_{p>\sqrt{M}}^{+\infty}\frac{\log p}{p}\chi(p)\right|\leq\frac{2}{\sqrt{M}}$$ and $$\sum_{p^{j}>M,j>2}^{+\infty}\frac{\log p}{p^{j/2}}\leq\left\{\begin{array}{crll} &\frac{\log M}{\sqrt{M}}&    \hbox{if}\ \ M+1 \hbox{ is prime,}\\ &\frac{1}{\sqrt{M}}&   \hbox{otherwise.}\end{array}\right.$$ Then $$\left\{\begin{array}{crll}|E_{M}|&\leq&\sqrt{\frac{n}{\log M}}\ \frac{\log M+2}{\sqrt{M}}\leq \sqrt{n}\ \frac{\log M+2}{\sqrt{M}}&  \hbox{if}\ M+1 \hbox{ is prime,}\\ |E_{M}|&\leq&\sqrt{\frac{n}{\log M}}\frac{3}{\sqrt{M}}\leq3\frac{\sqrt{n}}{\sqrt{M}}&\ \hbox{otherwise.}\end{array}\right.$$ Let $M$ be such that $|E_{M}|\leq 10^{-\nu}$. Then, using the theory of the Lambert $W$ function, we choose $M$ $$\left\{\begin{array}{crll}M&=&\frac{n}{4}\ \left[W_{-1}\left(-\frac{10^{-\nu}}{\sqrt{n}}\right)\right]^{2}+4n\ 10^{2\nu}& \hbox{if}\ M+1\ \hbox{ is prime,}\\ M&=&9n\ 10^{2\nu}&\hbox{otherwise,}\end{array}\right.$$ where $W_{-1}$ denotes the branch satisfying $W(x)\leq-1$ and $W(x)$ is the Lambert $W$ function, which is defined to be the multivalued inverse of the function $w\longmapsto we^{w}$.

[Partial Li criterion]{}\[sec.3\] In the following proposition, we propose a partial Li criterion which relates the partial Riemann hypothesis to the positivity of the Li coefficients up to a certain order.

\[prop.1\] For sufficiently large $T\geq T_{1}\geq1$, if every non-trivial zero $\rho$ of $L(s,\chi)$ with $|Im(\rho)|<T$ satisfies $Re(\rho)=1/2$, then $\lambda_{\chi}(n)\geq0$ for all $n\leq T^{2}$.

We have $$\lambda_{\chi}(n)=\sum_{\rho}\left[1-\left(1-\frac{1}{\rho}\right)^{n}\right].$$ Then $$\begin{aligned}
\label{eq.8}
\lambda_{\chi}(n)&=&\sum_{\rho}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)\nonumber\\
&=&\sum_{\rho; |Im(\rho)|<T}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)+\sum_{\rho; T<|Im(\rho)|}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)\nonumber\\\end{aligned}$$ Let $\rho=\beta+i\gamma$, then we obtain $$1-\left(1-\frac{1}{\rho}\right)^{n}=1-\left(\frac{1+\frac{\beta-1}{i\gamma}}{1+\frac{\beta}{i\gamma}}\right)^{n},\ \frac{1+\frac{\beta-1}{i\gamma}}{1+\frac{\beta}{i\gamma}}=\left(1-\frac{\beta}{\gamma^{2}}\right)+\frac{i}{\gamma}+O\left(\frac{1}{\gamma^{3}}\right)$$ and using binomial identity, we get $$Re\left[\left(\frac{1+\frac{\beta-1}{i\gamma}}{1+\frac{\beta}{i\gamma}}\right)^{n}\right]=\left(1-\frac{\beta}{\gamma^{2}}\right)^{n}+O\left(\frac{1}{\gamma^{3}}\right),$$ where the $O$-symbol depends on $n$. Therefore $$\begin{aligned}
\label{eq.9}\sum_{ |\gamma|>T}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)&=&\sum_{ |\gamma|>T}\left[1-\left(1-\frac{\beta}{\gamma^{2}}\right)^{n}+O\left(\frac{1}{\gamma^{3}}\right)\right]\nonumber\\
&=&\sum_{ |\gamma|>T}\frac{n\beta}{\gamma^{2}}+O\left(\frac{1}{\gamma^{3}}\right).\end{aligned}$$ Then, the second sum in goes to 0 in absolute value as $T \to \infty$ by conditional convergence (see also Proposition \[prop.2\] below). Finally, it suffices to prove that, under the Riemann hypothesis, there exists a positive constant $c_{0}$ such that for large $T$ we have $$\label{eq.11}
\sum_{|Im(\rho)|<T}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)\geq c_{0} \frac{\log T}{T}.$$ Thus, for $1\leq n\leq T^{2}$, we get $$\begin{aligned}
\label{eq.12}\sum_{ |\gamma|<T}\left(1-Re\left[\left(1-\frac{1}{\rho}\right)^{n}\right]\right)&\geq&\sum_{ |\gamma|<T}\frac{n^{2}}{2\gamma^{2}}\nonumber\\
&\geq&\frac{n^{2}}{2}\sum_{ |\gamma|<T}\frac{1}{\gamma^{2}}\nonumber\\
&\geq&\frac{n^{2}}{2T^{2}}\sum_{ |\gamma|<T}1\nonumber\\
&\geq&\frac{1}{2T^{2}}N_{\chi}(T).\end{aligned}$$ Recall that $$N_{\chi}(T)=\frac{1}{2\pi}T\log T+c_{1}T+O\left(\log T\right).$$ Then, equation is proved.

[**Remark**]{}

-   Recall that the $10^{13}$ first zeros of the Riemann zeta function lie on the line $Re(s)=1/2$ (see. [@4]). Then, from Proposition \[prop.1\], we might expect that the first $10^{26}$ Li coefficients $\lambda_{\zeta}(n)$ are non-negative.

-   In the section \[sec.5\], we will use the first $10^{4}$ critical zeros of the Dirichlet $L$-functions to compute the first Li coefficients $\lambda_{\chi}(n)$. Then, from Proposition \[prop.1\] above, we also might expect that the first $10^{8}$ Li coefficients are non-negative.

[**Conversely.**]{} From the work of Brown [@1], the first observation is that the first “non-trivial” inequality $\lambda_{\chi}(2)\geq0$ is sufficient to establish the non-existence of a Siegel zero for $\xi(s,\chi)$ (see [@1 Corollary 1]).\
  Let $r>1$ be a real number. By the invariance of the zeros $\rho$ of $\xi(s,\chi)$ under the map $\rho\longmapsto1-\overline{\rho}$, $$\forall \ \rho,\ \ \left|\frac{\rho}{\rho-1}\right|\leq r\ \Leftrightarrow\ \forall\ \rho,\ \rho\in{D_{r}},$$ where $D_{r}$ is the closed region bounded by the lines $\{z\in{\Bbb C}:\ Re(z)=0,1\}$ and the arcs of tow circles. The second observation (see [@1 Theorem 3]) is that, for large $N\in{\Bbb N}$, the equality $\lambda_{\chi}(1)\geq0,...,\lambda_{\chi}(n)\geq0$ imply the existence of a certain zero-free region for $\xi(s,\chi)$, that is, there exist constants $N,\mu, \nu$ depending only on $q$ such that if $\lambda_{\chi}(1)\geq0,...,\lambda_{\chi}(n)\geq0$ hold, and $n\geq N$, then the zeros of $\xi(s,\chi)$ belong to $D_{r}$, where $r=\sqrt{1+T^{-2}}$ and $T=\left(\frac{n}{\mu\log^{2}(\nu n)}\right)^{1/3}$.\
Let us define the partial Li coefficients by $$\lambda_{\chi}(n,T)=\sum_{\rho;\ |Im\rho|\leq T}1-\left(1-\frac{1}{\rho}\right)^{n}$$ with a parameter $T$. An estimate for the error term $|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)|$ is stated in the following proposition.

\[prop.2\] For sufficiently large $T\geq1$, we have $$\label{eq.12}
\left|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)\right|\leq\frac{3n^{2}}{2T^{2}}\left[\frac{1}{2\pi}T\log T+\left(\frac{1}{\pi}+\log\left(\frac{q}{2\pi e}\right)\right)T+\frac{1}{2}\right].$$

Note that $\rho=\beta+i\gamma$, where $\beta$, $\gamma\in{\rb}$ and $0\leq\beta\leq1$. We have $$\lambda_{\chi}(n)-\lambda_{\chi}(n,T)=\frac{1}{2}Re\left[ \sum_{|\gamma|> T}\left(2-\left(\frac{\rho-1}{\rho}\right)^{n}-\left(\frac{\rho}{\rho-1}\right)^{n}\right)\right].$$ Using a binomial expansion of the inner term in the sum in the right-hand side, we obtain $$\begin{aligned}
\label{eq.13}
Re\left[2-\left(\frac{\rho-1}{\rho}\right)^{n}-\left(\frac{\rho}{\rho-1}\right)^{n}\right]&=&Re\left[n\left(\frac{1}{\rho}+\frac{1}{1-\rho}\right)-\frac{n(n-1)}{2}\left(\frac{1}{\rho^{2}}+\frac{1}{(1-\rho)^{2}}\right)\right]\nonumber\\
&&+\ Re\left[\sum_{k=3}^{n}\left(_{k}^{n}\right)(-1)^{k-1}\left(\rho^{-k}+(1-\rho)^{-k}\right)\right].\end{aligned}$$ We have $$\frac{1}{1+\gamma^{2}}\leq Re\left(\frac{1}{\rho}+\frac{1}{1-\rho}\right)\leq\frac{1}{\gamma^{2}}$$ and $$\frac{1}{1+\gamma^{2}}-\frac{2}{\gamma^{4}}\leq Re\left(\frac{1}{\rho^{2}}+\frac{1}{(1-\rho)^{2}}\right)\leq\frac{2}{\gamma^{2}}.$$ Suppose now that $|\gamma|\geq T\geq n$, then $\frac{(n-3)}{|\gamma|}...\frac{(n-k)}{|\gamma|}\leq1$ for all $n\geq k\geq3$. Then $$\sum_{k=3}^{n}\left(_{k}^{n}\right)\frac{1}{|\gamma|^{k}}\leq2\frac{n^{3}}{|\gamma|^{3}}\sum_{k=3}^{\infty}\frac{1}{k!}=(2e-5)\frac{n^{3}}{|\gamma|^{3}}\leq\frac{n^{3}}{2|\gamma|^{3}}.$$ Therefore, $$\begin{aligned}
\label{eq.14}
Re\left[2-\left(\frac{\rho-1}{\rho}\right)^{n}-\left(\frac{\rho}{\rho-1}\right)^{n}\right]&\leq&\frac{n}{\gamma^{2}}+\frac{n^{2}-n}{\gamma^{2}}+\sum_{k=3}^{n}\left(_{k}^{n}\right)\frac{2}{|\gamma|^{k}}\nonumber\\
&\leq&\frac{n}{\gamma^{2}}+\frac{n^{3}}{2|\gamma|^{3}}\nonumber\\
&\leq&\frac{3n^{2}}{2|\gamma|^{2}}.\end{aligned}$$ Then $$\left|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)\right|\leq\frac{3}{4}n^{2}\sum_{|\gamma|> T}\frac{1}{\gamma^{2}}.$$ We have $$\label{eq.15}
\frac{1}{2}\sum_{|\gamma|>T}\frac{1}{\gamma^{2}}\leq\int_{T}^{\infty}-\frac{d}{dt}\left[t^{-2}\right]_{t=x}\left(N_{\chi}(x)-N_{\chi}(T)\right)dx
=\int_{T}^{\infty}x^{-2}dN_{\chi}(x).$$ Furthermore, $$\begin{aligned}
\int_{T}^{\infty}x^{-2}dN_{\chi}(x)&=&\int_{T}^{\infty}x^{-2}\left[\frac{1}{2\pi}\log x+\frac{1}{2\pi}+\log\left(\frac{q}{2\pi e}\right)+\frac{1}{x}\right]dx\nonumber\\
&=&T^{-2}\left[\frac{1}{2\pi}\log T+\frac{1}{2\pi}T\log T+\frac{1}{2\pi}T+\log\left(\frac{q}{2\pi e}\right)T+\frac{1}{2}\right].\nonumber\end{aligned}$$ Finally, we get $$\left|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)\right|\leq\frac{3}{2}\frac{n^{2}}{T^{2}}\left[\frac{1}{2\pi}T\log T+\left(\frac{1}{\pi}+\log\left(\frac{q}{2\pi e}\right)\right)T+\frac{1}{2}\right]$$ and Proposition \[prop.2\] follows.

For our computations at the end of this paper, we need to find $T_{0}$ such that $\left|\lambda_{\chi}(n)-\lambda_{\chi}(n,T)\right|\leq 10^{-k}$. To do so, it suffices to find $T_{0}$ such that $$\frac{3n^{2}}{4\pi}\frac{\log T}{T}\leq \frac{10^{-k}}{3}\ \ \ \Leftrightarrow \ \ \ \frac{\log T}{T}\leq \frac{4\pi 10^{-k}}{9n^{2}}.$$ Using the theory of the Lambert $W$ function, we get $$T_{0}=-\frac{9n^{2}}{4\pi}W_{-1}\left(-\frac{4\pi}{9n^{2}}10^{-k}\right),$$ where $W_{-1}$ denotes the branch satisfying $W(x)\leq-1$ and $W(x)$ is the Lambert $W$ function which is defined to be the multivalued inverse of the function $w\longmapsto we^{w}$.

[New formulas for the Li coefficients]{}\[sec.4\] In this section, we give new formulas for the Li coefficients under the Riemann hypothesis (integral and summation formula) which will be used to compute and verify the positivity of the Li coefficients $\lambda_{\chi}(n)$ under the Riemann hypothesis.\
From (\[eq.1\]) we have $$\label{eq.16}
\log\xi\left(\frac{z}{z-1},\chi\right)=\log\xi\left(\frac{1}{1-z},\chi\right)=\log \xi(0,\chi)+ \sum_{n=1}^{\infty}\lambda_{\chi}(n)\frac{z^{n}}{n}.$$ The number $\lambda_{\chi}(n)$ does not depend on the choice of the logarithm. Rewrite (\[eq.16\]) at the point $z=-1$. Note that the region of convergence for this is an open disk of radius 2 centered at $z=-1$ and it encloses in particular the entirety of the closed unit disk, except for the point $z=1$ that is a pole of $\xi(\frac{1}{1-z},\chi)$.\
Assume that the Riemann hypothesis holds. Then, we have $$\begin{aligned}
\label{eq.17}
\log\xi\left(\frac{1}{1-z},\chi\right)&=&\log \xi(0,\chi)+ \sum_{n=1}^{\infty}\lambda_{\chi}(n)\frac{z^{n}}{n}\nonumber\\
&=&C_{\chi}(0)+\sum_{n=1}^{\infty}C_{\chi}(n)(z+1)^{n}.\end{aligned}$$ Expanding $(z+1)^{n}$, we obtain $$\label{eq.18}
\lambda_{\chi}(n)=n\sum_{j=1}^{\infty}\left(_{n}^{j}\right)C_{\chi}(j).$$ We have $$\label{eq.19}
N_{\chi}(T)=\frac{1}{\pi}Im\left(\log\xi_{\chi}\left(\frac{1}{2}+iT\right)\right)=\sum_{n=1}^{\infty}\frac{C_{\chi}(n)}{\pi}\left(Im\left(\frac{2\gamma+i}{2\gamma-i}+1\right)\right)^{n},$$ where we have used the substitution $\frac{1}{2}+i\gamma=\frac{1}{1-z}$ or $z=\frac{2\gamma+i}{2\gamma-i}$. Then, since $$\left(Im\left(\frac{2\gamma+i}{2\gamma-i}+1\right)\right)^{n}=\frac{(4\gamma)^{n}}{(4\gamma^{2}+1)^{n/2}}\sin\left(n\tan^{-1}\frac{1}{2\gamma}\right)=2^{n}\cos^{n}\theta\sin(n\theta),$$ $$\cos\theta:=\frac{2\gamma}{\sqrt{4\gamma^{2}+1}},$$ we get $$\pi  N_{\chi}(\gamma)=\sum_{n=1}^{\infty}C_{\chi}(n)2^{n}\cos^{n}\theta\sin(n\theta).$$ Using the identity $$\int_{0}^{\pi/2}\cos^{n}\theta\sin(n\theta)\sin(2m\theta)d\theta=\frac{\pi}{2^{n+2}}\left(_{m}^{n}\right),\ \ m, n\in{\nb},$$ we deduce $$\int_{0}^{\pi/2}\pi N_{\chi}(\gamma)\sin(2m\theta)d\theta=\sum_{n=1}^{\infty}C_{\chi}(n)\frac{\pi}{4}\left(_{m}^{n}\right).$$ Hence, $$\sum_{n=1}^{\infty}C_{\chi}(n)\left(_{m}^{n}\right)=4\int_{0}^{\pi/2}N_{\chi}(\gamma)U_{m-1}(\cos(2\theta))\sin(2\theta)d\theta,$$ where $U_{m-1}$ are the Chebyschev polynomial of the second kind. Using that $$U_{m-1}(\cos\theta):=\frac{\sin(m\theta)}{\sin(\theta)},\ \ \ \cos(2\theta)=\cos^{2}\theta-\sin^{2}\theta=\frac{4\gamma^{2}-1}{4\gamma^{2}+1},$$ $$\sin(2\theta)d\theta=-2\cos\theta d(\cos\theta)$$ and that as $\gamma$ proceeds from 0 to $\infty$, $\theta$ subtends an angle from $\pi/2$ to $0$, we obtain $$\sum_{n=1}^{\infty}C_{\chi}(n)\left(_{m}^{n}\right)=8\int_{0}^{\infty}N_{\chi}(\gamma)U_{m-1}\left(\frac{4\gamma^{2}-1}{4\gamma^{2}+1}\right)\times\frac{2\gamma}{\sqrt{4\gamma^{2}+1}}\times\frac{2}{(4\gamma^{2}+1)^{3/2}}d\gamma.$$ Therefore, from (\[eq.18\]) we get for all $n\in{\nb}$ the following proposition.

\[prop.3\] Under the Riemann hypothesis, for $n\geq1$, we have $$\label{eq.20}
\lambda_{\chi}(n)=32\ n\ \int_{0}^{\infty}\frac{\gamma}{(4\gamma^{2}+1)^{2}}N_{\chi}(\gamma)U_{n-1}\left(\frac{4\gamma^{2}-1}{4\gamma^{2}+1}\right)d\gamma.$$

Next, we give another formula for the Li coefficient. Recall that the function $N_{\chi}(T)$ is a real step function, increasing by unity each time a new critical zero is counted: $$\label{eq.21}N_{\chi}(T)=\sum_{\rho, Im(\rho)>0}\phi(T-Im(\rho))=\sum_{k=1}^{\infty}\alpha_{k}\phi(T-\gamma_{k}),$$ where $\rho_{j}=\beta_{k}+i\gamma_{k},\ \gamma_{k}>0$ and $\phi(x-a)=1$ if $x\geq a$ and 0 if $x<a$. The zeros are ordered so that $\gamma_{k+1}>\gamma_{k}$ and the $\alpha_{k}$ counts the number of zeros with imaginary part $\gamma_{k}$ including the multiplicities. Simplification of the integral formula (\[eq.20\]) is stated in the following proposition.

\[prop.4\] Under the Riemann hypothesis, we have $$\lambda_{\chi}(n)=2\sum_{k=1}^{\infty}\alpha_{k}\left(1-T_{n}\left(\frac{4\gamma_{k}^{2}-1}{4\gamma_{k}^{2}+1}\right)\right),\ \ n\in{\nb}.$$

By (\[eq.21\]), the formula (\[eq.20\]) can be written as follows: $$\begin{aligned}
  \lambda_{\chi}(n)&=&32n\sum_{k=1}^{\infty}\alpha_{k}\int_{0}^{\infty}\phi(\gamma-\gamma_{k})\frac{\gamma}{(4\gamma^{2}+1)^{2}}U_{n-1}\left(\frac{4\gamma^{2}-1}{4\gamma^{2}+1}\right)d\gamma\nonumber\\
&=&2n\sum_{k=1}^{\infty}\alpha_{k}\int_{\gamma_{k}}^{\infty}\frac{16\gamma}{(4\gamma^{2}+1)^{2}}U_{n-1}\left(\frac{4\gamma^{2}-1}{4\gamma^{2}+1}\right)d\gamma\nonumber\\
&=&2n\sum_{k=1}^{\infty}\alpha_{k}\left[\frac{1}{n}T_{n}(y)\right]_{\frac{4\gamma_{k}^{2}-1}{4\gamma_{k}^{2}+1}}^{1}\nonumber\\
&=&2\sum_{k=1}^{\infty}\alpha_{k}\left(1-T_{n}\left(\frac{4\gamma_{k}^{2}-1}{4\gamma_{k}^{2}+1}\right)\right),\nonumber\end{aligned}$$ using the following relation between the Chebyshev polynomials of the second kind and the first kind $$\int U_{n}(x)dx=\frac{1}{n+1}T_{n+1}(x).$$

This is remarkable summation expression for the Li coefficients. We numerically evaluate some of the first terms by the right hand side expression and find them to be indeed close to the required values of the Li coefficients. This is reassuring, and the results are presented in the tables below.\
Under the Riemann hypothesis, from the above arguments used in the proof of Propositions \[prop.3\] and \[prop.4\], one can derive the following formula $$\lambda_{\chi}(n,T)=2\sum_{k=1}^{N}\alpha_{k}\left(1-T_{n}\left(\frac{4\gamma_{k}^{2}-1}{4\gamma_{k}^{2}+1}\right)\right),$$ where $N=[N_{\chi}(T)]$ with $[x]=x-\{x\}$ and $\{x\}$ denotes the fractional part of $x$ (the last formula will be denoted $\lambda_{\chi}(n,N)$). Therefore, the latter formula allows one to estimate the error term $|\lambda_{\chi}(n)-\lambda_{\chi}(n,N)|$ in Proposition \[prop.4\] by evaluating directly the partial Li coefficients as in Proposition \[prop.2\].

[Numerical computations]{}\[sec.5\]

In this section, we compute and verify the positivity of the values of $\lambda_{\chi}(n)$ unconditionally or under the Riemann hypothesis. We first compute unconditionally (without assuming the Riemann hypothesis) $\tilde{\lambda}_{\chi}(n,M)$ by using equation (\[eq.3\]) and computing prime numbers up to $M$ (see. Section \[sec.2\]). We also compute under the Riemann hypothesis $$\label{eq.22}\lambda_{\chi}(n,N)=2\sum_{k=1}^{N}\alpha_{k}\left(1-T_{n}\left(\frac{4\gamma_{k}^{2}-1}{4\gamma_{k}^{2}+1}\right)\right),\ \hbox{with}\  N=10^{4}.$$ Furthermore, we carried out the calculations for several examples of characters. Some illustrative examples are cited below. We restricted the tables below for $n\leq 40$. However, one can find the other values of $n>40$ represented in the graphs 1-4.\
[**Remark.**]{} In fact, by the summation formula (\[eq.22\]), we could compute more coefficients $\lambda_{\chi}(n)$ with less time consuming way than by the arithmetic formula (\[eq.3\]), where computation of the first 50 coefficients lasted more than a week.\
Based on the tables below, we conjecture the following result.\
[**Conjecture. The coefficients $\lambda_{\chi}(n)$ are positive and increasing in $n$.**]{}\
This conjecture was partially numerical verified for the case of the Riemann zeta function (see [@2 Appendix D] and [@8]) and by the authors in a work in progress for the Hecke $L$-functions [@13].

  -------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
  $\chi$(mod3)                                                                                                                                            
  $n$              $\tilde{\lambda}_{\chi}(n,M)\  $    $\lambda_{\chi}(n,N)$       $n$       $\tilde{\lambda}_{\chi}(n,M)$     $\lambda_{\chi}(n,N)$      
  1                            0.05316                       0.056442              19                  17.18050                      17.16170             
  2                            0.22763                        0.22542              20                  18.58480                      18.69100             
  3                            0.14844                        0.50592              21                  20.01400                      20.24310             
  4                            0.89344                        0.89624              22                  21.46700                      21.81300             
  5                            1.35725                        1.39404              23                  22.94280                      23.39600             
  6                            2.12951                        1.99635              24                  24.44030                      24.98820             
  7                            2.98573                        2.69962              25                  25.95870                      26.58590             
  8                            3.91334                        3.49978              26                  27.49700                      28.18600             
  9                            4.40970                        4.39225              27                  29.05460                      29.78580             
  10                           5.94841                        5.37202              28                  30.63070                      31.38330             
  11                           7.04344                        6.43371              29                  32.22460                      32.97700             
  12                           8.18382                        7.57163              30                  33.83580                      34.56580             
  13                           9.36580                        8.77987              31                  35.46370                      36.14940             
  14                          10.58620                       10.05230              32                  37.10770                      37.72780             
  15                          11.84230                       11.38280              33                  38.76730                       39.3014             
  16                          12.81150                       12.76510              34                  40.44210                      40.87120             
  17                          14.45250                       14.19300              35                  42.13150                      42.43870             
  18                          15.80260                       15.66050              36                  43.83530                      44.00550             
  -------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --

![ Case of $\chi$ (mod3)[]{data-label="fig:1"}](graphmod3){height="8cm"}

  -------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
  $\chi$(mod5)                                                                                                                                            
  $n$              $\tilde{\lambda}_{\chi}(n,M)\  $    $\lambda_{\chi}(n,N)$       $n$       $\tilde{\lambda}_{\chi}(n,M)$     $\lambda_{\chi}(n,N)$      
  1                            0.13183                        0.08562              21                  25.37770                      26.21450             
  2                            0.29872                        0.34152              22                  27.08610                      27.92160             
  3                            0.91468                        0.76482              23                  28.81730                      29.60960             
  4                            1.58476                        1.35081              24                  30.57020                      31.27780             
  5                            2.63432                        2.09300              25                  32.34400                      32.92720             
  6                            3.66199                        2.98332              26                  34.13770                      34.56020             
  7                            4.77362                        4.01225              27                  35.95070                      36.18030             
  8                            5.95664                        5.16902              28                  37.78220                      37.79200             
  9                            7.06010                        6.44188              29                  39.63160                      39.40090             
  10                           8.50254                        7.81828              30                  41.49820                      41.01320             
  11                           9.85298                        9.28519              31                  43.38150                      42.63540             
  12                          11.24880                       10.82930              32                  45.28090                       44.2746             
  13                          12.68620                       12.43740              33                  47.19590                      45.93760             
  14                          14.16200                       14.09650              34                  49.12610                      47.63100             
  15                          15.67350                       15.79410              35                  51.07100                      49.36130             
  16                          17.68370                       17.51860              36                  53.03020                      51.13410             
  17                          14.45250                       19.25930              37                  55.00320                      52.95430             
  18                          20.40000                       21.00670              38                  56.98980                      54.82600             
  19                          22.03340                       22.75260              39                  58.98950                      56.75210             
  20                          23.69300                       24.49030              40                  61.00210                      58.73450             
  -------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --

![Case of $\chi$ (mod5)[]{data-label="fig:1"}](graphmod5){height="7cm"}

  --------------- --------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
  $\chi$(mod20)                                                                                                                                          
  $n$               $\tilde{\lambda}_{\chi}(n,M)$     $\lambda_{\chi}(n,N)$       $n$       $\tilde{\lambda}_{\chi}(n,M)$     $\lambda_{\chi}(n,N)$      
  1                           0.695021                      0.319128              21                  39.93370                      41.70260             
  2                            1.68502                       1.24419              22                  42.33530                      44.31350             
  3                            2.99412                       2.68343              23                  44.75970                      46.59570             
  4                            4.48123                       4.50032              24                  47.20580                      48.55720             
  5                            6.10005                       6.53527              25                  49.67270                      50.26430             
  6                            7.82087                       8.63067              26                  52.15960                      51.83150             
  7                            9.62565                      10.65500              27                  54.66570                      53.403100            
  8                           11.50180                      12.52230              28                  57.19040                      55.12930             
  9                           13.32220                      14.20280              29                  59.73290                      57.14130             
  10                          15.43400                      15.72450              30                  62.29260                      59.52940             
  11                          17.47760                      17.16450              31                  64.86910                      62.32740             
  12                          19.56650                      18.63130              32                  67.46160                      65.50710             
  13                          21.69710                      20.24300              33                  70.06980                      68.98220             
  14                          23.86610                      22.10320              34                  72.69310                      72.62260             
  15                          26.07070                      24.28030              35                  75.33110                      76.27560             
  16                          28.54690                      26.79300              36                  77.98350                      79.79060             
  17                          30.57800                      29.60520              37                  80.64970                      83.04340             
  18                          32.87670                      32.63050              38                  83.32940                      85.95580             
  19                          35.20320                      35.74610              39                  86.02230                      88.50750             
  20                          37.55600                      38.81360              40                  88.72800                      90.73760             
  --------------- --------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --

![Case of $\chi$ (mod20)[]{data-label="fig:1"}](graphmod20){height="7cm"}

  --------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --
  $\chi$(mod60)                                                                                                                                            
  $n$               $\tilde{\lambda}_{\chi}(n,M)\  $    $\lambda_{\chi}(n,N)$       $n$       $\tilde{\lambda}_{\chi}(n,M)$     $\lambda_{\chi}(n,N)$      
  1                             1.12226                        0.48626              21                  51.46920                      50.88960             
  2                             2.78363                        1.86950              22                  54.42010                      52.52830             
  3                             4.64204                        3.94169              23                  57.39370                      54.44350             
  4                             6.83662                        6.41363              24                  60.38910                      56.86290             
  5                             8.84658                        8.98530              25                  63.40530                      59.89590             
  6                            11.11670                       11.41720              26                  66.44150                      63.50750             
  7                            13.47080                       13.58380              27                  69.49700                      67.53000             
  8                            15.89630                       15.49640              28                  72.57090                      71.70750             
  9                            18.06830                       17.28820              29                  75.6628                        75.7637             
  10                           20.92710                       19.16770              30                  78.77180                      79.47310             
  11                           23.52000                       21.35250              31                  81.89750                      82.71770             
  12                           26.15820                       24.00100              32                  85.03940                      85.51520             
  13                           28.83810                       27.16160              33                  88.19690                      88.00960             
  14                           31.55630                       30.75170              34                  91.36950                      90.42920             
  15                           34.31030                       34.57380              35                  94.55690                      93.02160             
  16                           37.56690                       38.36300              36                  97.75850                      95.98430             
  17                           39.91620                       41.85530              37                 100.97400                      99.40850             
  18                           42.76420                       44.85610              38                 104.20300                      103.25300            
  19                           45.64000                       47.29300              39                 107.44500                      107.35200            
  20                           48.54210                       49.23760              40                 110.70000                      111.46000            
  --------------- ----------------------------------- ------------------------- ---------- ---------------------------------- ------------------------- -- --

![Case of $\chi$ (mod60)[]{data-label="fig:1"}](graphmod60){height="7cm"}

[**Acknowledgements**]{}. The authors would like to thank Maciej Radziejewski and Mark Coffey for their many valuable comments about the published paper.

[99]{}

, [*Li’s criterion and zero-free regions of $L$-functions,*]{} Journal of Number Theory [**111**]{} (2005) 1-–32. , [ *Toward verification of the Riemann hypothesis: Application of the Li criterion,*]{} Math. Phys. Anal. Geom. [**8**]{} (3) (2005) 211–255. , [*Multiplicative Number Theory,*]{} Springer-Verlag, New York, (1980). , [*The $10^{13}$ first zeros of the Riemann zeta function, and zeros computation at very large height,*]{} available at http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf, October 2004. , [*Bounded nonvanishing functions and Bateman functions,*]{} Complex Variables [**25**]{} (1994) 237–259. , [*The positivity of a sequence of numbers and the Riemann hypothesis,*]{} J. Number theory [**65**]{} (2) (1997) 325–333. , [*Explicit formulas for Dirichlet and Hecke $L$-functions,*]{} Illinois J. Math. [**48**]{} (2) (2004) 491–503. , [*Li’s criterion for the Riemann hypothesis - numerical approch,*]{} Opuscula Mathematica [**24**]{} (1) (2004) 103–114. , [*Li’s criterion and the Riemann hypothesis for function fields,*]{} Finite Fields and Their Applications [**16**]{} (6) (2010) 477–485. , [*Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg,*]{} J. Number Theory [**125**]{} (1) (2007) 50–58. , [*Erratum et addendum à l’article, “Le critère de Li et l’hypothèse de Riemann pour la classe de Selberg” \[J. Number Theory [**125**]{} (2007) 50–58\],* ]{} J. Number Theory [**130**]{} (4) (2010) 1098–1108. , [*The Li criterion and the Riemann hypothesis for the Selberg class II*]{}, J. Number Theory [**130**]{} (4) (2010) 1109–1114. , [*On the zeros of Hecke $L$-functions,*]{} Preprint, 2011.

[^1]: 2000 Mathematics Subject Classification: 11M06,11M26, 11M36.\
    Keywords and Phrases: Dirichlet $L$-functions, Li’s criterion, Riemann hypothesis.
