arXiv subjects
Publications and source records attributed to Kamel Mazhouda.
In this paper, new asymptotic formulas for sums over zeros of functions from the Selberg class are obtained. These results continue the investigations of Murty $\&$ Perelli \cite{12}, of Murty $\&$ Zaharescu \cite{13}, of Kamiya $\&$ Suzuki \cite{8}, of Steuding \cite{19} and other authors.
In this paper, we study the value distribution of the derivative of a Dirichlet $L$-function $L'(s,χ)$ at the $a$-points $ρ_{a,χ}=β_{a,χ}+iγ_{a,χ}$ of $L(s,χ).$ We give an asymptotic formula for the sum $$\sum_{ρ_{a,χ};\ 0<γ_{a,χ}\leq T}L'\left(ρ_{a,χ},χ\right) X^{ρ_{a,χ}}\ \ \hbox{as}\ \ T\longrightarrow \infty,$$ where $X$ is a fixed positive number and $χ$ is a primitive character $\mod q$. This work continues the investigations of Fujii \cite{2,3,4}, Garunk$\check{s}$tis \& Steuding \cite{7} and the authors \cite{12}.
In this paper we prove that the Generalized Riemann Hypothesis (GRH) for functions in the class $\mathcal{S}^{\sharp\flat}$ containing the Selberg class is equivalent to a certain integral expression of the real part of the generalized Li coefficient $λ_F(n)$ associated to $F\in\mathcal{S}^{\sharp\flat}$, for positive integers $n$. Moreover, we deduce that the GRH is equivalent to a certain expression of $Re(λ_F(n))$ in terms of the sum of the Chebyshev polynomials of the first kind. Then, we partially evaluate the integral expression and deduce further relations equivalent to the GRH involving the generalized Euler-Stieltjes constants of the second kind associated to $F$. The class $\mathcal{S}^{\sharp\flat}$ unconditionally contains all automorphic $L$-functions attached to irreducible cuspidal unitary representations of $GL_N(\mathbb{Q})$, hence, as a corollary we also derive relations equivalent to the GRH for automorphic $L$-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 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.
In this paper, we apply the saddle-point method in conjunction with the theory of the N$\ddot{o}$rlund-Rice integrals to derive a precise asymptotic formula for the generalized Li coefficients established by Omar and Mazhouda. Actually, for any function $F$ in the Selberg class $\mathcal{S}$ and under the Generalized Riemann Hypothesis, we have $$λ_{F}(n)=\frac{d_{F}}{2}n\log n+c_{F}n+O(\sqrt{n}\log n),$$ with $$c_{F}=\frac{d_{F}}{2}(γ-1)+\frac{1}{2}\log(λQ_{F}^{2}),\ \ λ=\prod_{j=1}^{r}λ_{j}^{2λ_{j}},$$ where $γ$ is the Euler constant and the notation is as bellow.
Let $F$ be a function in the Selberg class ${\mathcal S}$ and $a$ be a real number not equal to 1/2. Consider the sum $$λ_{F}(n,a)=\sum_ρ\left[1-\left(\frac{ρ-a}{ρ+a-1}\right)^{n}\right],$$ where $ρ$ runs over the non-trivial zeros of $F$. In this paper, we prove that the Riemann hypothesis is equivalent to the positivity of the "modified Li coefficient" $λ_{F}(n,a)$, for $n=1,2,..$ and $a<1/2$. Furthermore, we give an explicit arithmetic and asymptotic formula of these coefficients.