Second moment of the Beurling zeta-function

In this paper we obtain a mean value theorem for a general Dirichlet series $f(s)= \sum_{j=1}^\infty a_j n_j^{-s}$ with positive coefficients for which the counting function $A(x) = \sum_{n_{j}\le x}a_{j}$ satisfies $A(x)=\rho x + O(x^\beta)$ for some ρ > 0 and β < 1. We prove that $\frac1T\int_0^T |\,f(\sigma+it)|^2\, dt \to \sum_{j=1}^\infty a_j^2n_j^{-2\sigma}$ for $\sigma\gt\frac{1+\beta}{2}$ and obtain an upper bound for this moment for $\beta\lt\sigma\le \frac{1+\beta}{2}$. We provide a number of examples indicating the sharpness of our results.

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On Universality of Some Beurling Zeta-Functions

Geštautas,

Laurinčikas

2024

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Let P be the set of generalized prime numbers, and ζP(s), s=σ+it, denote the Beurling zeta-function associated with P. In the paper, we consider the approximation of analytic functions by using shifts ζP(s+iτ), τ∈R. We assume the classical axioms for the number of generalized integers and the mean of the generalized von Mangoldt function, the linear independence of the set {logp:p∈P}, and the existence of a bounded mean square for ζP(s). Under the above hypotheses, we obtain the universality of the function ζP(s). This means that the set of shifts ζP(s+iτ) approximating a given analytic function defined on a certain strip σ^<σ<1 has a positive lower density. This result opens a new chapter in the theory of Beurling zeta functions. Moreover, it supports the Linnik–Ibragimov conjecture on the universality of Dirichlet series.For the proof, a probabilistic approach is applied.

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Second moment of the Beurling zeta-function

Cited by 4 publications

References 13 publications

On Second Moment of Selberg Zeta-Function for $$\sigma =1$$

On Second Moment of Selberg Zeta-Function for $$\sigma =1$$

A Mean Value Theorem for General Dirichlet Series

On Universality of Some Beurling Zeta-Functions

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