Related papers: Selberg's Conjectures and Artin $L$-functions

The analytic properties of the standard twist $F(s,\alpha)$, where $F(s)$ belongs to a wide class of $L$-functions, are of prime importance in describing the structure of the Selberg class. In this paper we present a deeper study of such…

The paper proves a generalization of Wintner's theorem on the asymptotics of summation functions to the case of summation functions with nonlinear asymptotics. The class of arithmetic functions that have a logarithmic asymptotic mean is…

The secondary zeta function $Z(s)=\sum_{n=1}^\infty\alpha_n^{-s}$, where $\rho_n=\frac12+i\alpha_n$ are the zeros of zeta with $\Im(\rho)>0$, extends to a meromorphic function on the hole complex plane. If we assume the Riemann hypothesis…

In this article, we study the density conjecture of Katz and Sarnak for $L$-functions of ad\'elic Hilbert modular forms and their convolutions. In particular, under the generalised Riemann hypothesis, we establish several instances…

These notes were written from a series of lectures given in March 2010 at the Universidad Complutense of Madrid and then in Barcelona for the centennial anniversary of the Spanish Mathematical Society (RSME). Our aim is to give an…

Using the ideas of Deninger, we prove that the Artin $L$-functions coincide with such of the noncommutative tori. This result can be viewed as the Langlands reciprocity for noncommutative tori.

We first briefly survey the value-distribution theory of L-functions of the Bohr-Jessen flavor (or the theory of "M-functions"). Limit formulas for the Riemann zeta-function, Dirichlet L-functions, automorphic L-functions etc. are…

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…

This work is dedicated to the promotion of the results Hadamard, Landau E., Walvis A., Estarmann T and Paul R. Chernoff for pseudo zeta functions. The properties of zeta functions are studied, these properties can lead to new regularities…

We prove a version of the Extra-zero conjecture formulated by the first named author for p-adic L-functions associated to Rankin-Selberg convolutions of modular forms of the same weight. The novelty of this result is to provide strong…

This is a survey article on the twisted dynamical zeta functions of Ruelle and Selberg and the Fried's conjecture. It is based on the mini-course: "Twisted Ruelle zeta function, complex-valued analytic torsion and the Fried's conjecture",…

We prove the universality theorem for the iterated integrals of logarithms of $L$-functions in the Selberg class on some line parallel to the real axis.

Let $G$ be a finite group and $A$ be a regular local ring on which $G$ acts. Under certain assumptions on $A$ and the action, Serre defined a function $a_G\colon G\rightarrow\mathbb{Z}$ which can be viewed as a higher dimensional analogue…

Let $1/2\leq\beta<1$, $p$ be a generic prime number and $f_\beta$ be a random multiplicative function supported on the squarefree integers such that $(f_\beta(p))_{p}$ is an i.i.d. sequence of random variables with distribution…

We formulate (and prove under a certain assumption) a conjecture relating the order of Somekawa's Milnor $K$-group attached to a torus $T$ and the value of the Artin $L$-function attached to the cocharacter group of $T$ (regarded as an…

A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist--Meurman--type congruences for the universal Bernoulli polynomials that are related…

In this essay I will give a strictly subjective selection of different types of zeta functions. Instead of providing a complete list, I will rather try to give the central concepts and ideas underlying the theory. This article is going to…

The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with…

For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of…

In 2023, Li, Du, Yi proved a uniqueness theorem for L functions in the extended Selberg class under the assumptions of positive degree, a shared functional equation, and the sharing of three complex values. This was later strengthened by…