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On universality of general Dirichlet series

Frédéric Bayart, Athanasios Kouroupis

Abstract

In the present work, we establish sufficient conditions for a Dirichlet series induced by general frequencies to be universal with respect to vertical translations. Our results can be applied to known universal objects such as Hurwitz zeta functions and also can provide new examples of universal Dirichlet series including the alternating prime zeta function $\sum_{n\geq1}(-1)^np_n^{-s}$.

On universality of general Dirichlet series

Abstract

In the present work, we establish sufficient conditions for a Dirichlet series induced by general frequencies to be universal with respect to vertical translations. Our results can be applied to known universal objects such as Hurwitz zeta functions and also can provide new examples of universal Dirichlet series including the alternating prime zeta function .
Paper Structure (14 sections, 25 theorems, 107 equations, 1 figure)

This paper contains 14 sections, 25 theorems, 107 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abscissas of convergence
  4. How to prove universality
  5. Two lemmas to estimate exponential sums
  6. The incomplete Gamma function/ Prym's function
  7. A remark on Bayrearrangement
  8. Landau's Theorem revisited
  9. Conditions (LC) and (WLC)
  10. Proof of Landau's theorem
  11. Application to the square moments of the zeta function
  12. Examples of strongly universal convergent Dirichlet series
  13. Proof of Theorem \ref{['thm:main2']}
  14. Random models and further discussion

Key Result

Theorem 1.1

Let $(\lambda_n)$ be a frequency, let $D(s)=\sum_{n\geq 1}a_n e^{-\lambda_n s}$ be a Dirichlet series. Assume that the frequency $(\lambda_n)$ is $\mathbb Q$-linearly independent, satisfies (WLC) and that for all $\alpha,\,\beta>0,$ there exist $C>0$ and $x_0\geq 1$ such that, for all $x\geq x_0,$ Then $D$ is strongly universal in $\{(\sigma_c(D)+\sigma_a(D))/2<\Re e(s)<\sigma_a(D)\}$.

Figures (1)

  • Figure :

Theorems & Definitions (46)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 36 more