Related papers: On relations equivalent to the generalized Riemann…

We formulate a generalization of Riesz-type criteria in the setting of $L$-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the Grand Riemann Hypothesis (GRH) for $L$-functions satisfying axioms of the…

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 $$\lambda_{F}(n,a)=\sum_{\rho}\left[1-\left(\frac{\rho-a}{\rho+a-1}\right)^{n}\right],$$ where $\rho$ runs over the…

We define generalized Li coefficients, called $\tau-$Li coefficients for a very broad class $\mathcal{S}^{\sharp \flat }(\sigma_0, \sigma_1)$ of $L-$functions that contains the Selberg class, the class of all automorphic $L-$functions and…

A short proof of the generalized Riemann hypothesis (gRH in short) for zeta functions $\zeta_{k}$ of algebraic number fields $k$ - based on the Hecke's proof of the functional equation for $\zeta_{k}$ and the method of the proof of the…

Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds for moduli of $\log{\mathcal{L}(s)}$ and $\mathcal{L}'(s)/\mathcal{L}(s)$ in the neighbourhood of the 1-line when $\mathcal{L}(s)$ are the Riemann, Dirichlet and…

We show that the Generalized Riemann Hypothesis for all Dirichlet L-functions is a consequence of certain conjectural properties of the zeros of the Riemann zeta function. Conversely, we prove that the zeros of $\zeta(s)$ satisfy those…

For each primitive Dirichlet character $\chi$, a hypothesis ${\rm GRH}^\dagger[\chi]$ is formulated in terms of zeros of the associated $L$-function $L(s,\chi)$. It is shown that for any such character, ${\rm GRH}^\dagger[\chi]$ is…

The standard twist $F(s,\alpha)$ of $L$-functions $F(s)$ in the Selberg class has several interesting properties and plays a central role in the Selberg class theory. It is therefore natural to study its finer analytic properties, for…

In the present paper, we prove that the generalized Riemann hypothesis for the Dirichlet $L$-function $L(s,\chi)$ is equivalent to the following bound: Let $k \geq 1$ and $\ell$ be positive real numbers. For any $\epsilon >0$, we have…

Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients lambda_n, indexed by the integers. We define similar coefficients attached to principal automorphic L-functions over GL(N). We…

We give a full description of the functions $F$ of degree 2 and conductor 1 in the general framework of the extended Selberg class. This is performed by means of a new numerical invariant $\chi_F$, which is easily computed from the data of…

In this note we define L-functions of finite graphs and study the particular case of finite cycles in the spirit of a previous paper that studied spectral zeta functions of graphs. The main result is a suggestive equivalence between an…

Assuming the Generalized Riemann Hypothesis, we provide uniform upper and lower bounds with explicit main terms for $\log{\left|\cL(s)\right|}$ for $\sigma \in (1/2,1)$ and for functions in the Selberg class. In particular, we focus on the…

We show that central zeros of $L$-functions in the Selberg class have a probabilistic interpretation by stating an equivalence condition of the Riemann hypothesis for the $L$-functions in terms of infinitely divisible distributions.

In 1916, Riesz gave an equivalent criterion for the Riemann hypothesis (RH). Inspired from Riesz's criterion, Hardy and Littlewood showed that RH is equivalent to the following bound: \begin{align*} P_1(x):= \sum_{n=1}^\infty…

We introduce screw functions for Dirichlet series in the extended Selberg class. Then we prove that the Grand Riemann Hypothesis for a member of the extended Selberg class is equivalent to the nonpositivity of the corresponding screw…

We prove Hypothesis H in full generality for ${\rm GL}_n$ over any number field. This result is a consequence of our stronger effective bound on Euler products involving Rankin--Selberg coefficients at prime ideal powers. The proof rests on…

Let $k$ be a number field and $G$ be a finite group. Let $\mathfrak{F}_{k}^{G}(Q)$ be the family of number fields $K$ with absolute discriminant $D_K$ at most $Q$ such that $K/k$ is normal with Galois group isomorphic to $G$. If $G$ is the…

Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of $\left(\cL'/\cL\right)(s)$ and $\log{\cL(s)}$ for $1/2+\delta\leq\sigma<1$, fixed $\delta\in(0,1/2)$ and for functions in…

We propose analogs of the classical Generalized Riemann Hypothesis and the Generalized Simplicity Conjecture for the characteristic p L-series associated to function fields over a finite field. These analogs are based on the use of absolute…