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Limit theorem for the hybrid joint universality theorem on zeta and $L$-functions

Kenta Endo

Abstract

In 1979, Gonek presented the hybrid joint universality theorem for Dirichlet $L$-functions and proved the universality theorem for Hurwitz zeta-functions with rational parameter as an application. Following the introduction of the hybrid universality theorem, several generalizations, refinements, and applications have been developed. Despite these advancements, no probabilistic proof based on Bagchi's approach has been formulated due to the complexities of adapting his method to the hybrid joint universality theorem. In this paper, we prove the limit theorem for the hybrid joint universality theorem.

Limit theorem for the hybrid joint universality theorem on zeta and $L$-functions

Abstract

In 1979, Gonek presented the hybrid joint universality theorem for Dirichlet -functions and proved the universality theorem for Hurwitz zeta-functions with rational parameter as an application. Following the introduction of the hybrid universality theorem, several generalizations, refinements, and applications have been developed. Despite these advancements, no probabilistic proof based on Bagchi's approach has been formulated due to the complexities of adapting his method to the hybrid joint universality theorem. In this paper, we prove the limit theorem for the hybrid joint universality theorem.

Paper Structure

This paper contains 5 sections, 15 theorems, 122 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main result
  3. Preliminaries
  4. Proof of Theorem \ref{['thm:main']} (i)
  5. Proof of (ii) of Theorem \ref{['thm:main']}

Key Result

Theorem 1.1

Let $K$ be a compact subset of $\{ s \in \mathbb{C} ~;~ 1/2 < \mathrm{Re}\space (s) < 1 \}$ with connected complement, $\chi_1,\, \ldots,\, \chi_r$ be pairwise non-equivalent Dirichlet characters, and $p_{k_1}, \, \ldots,\, p_{k_N}$ be mutually distinct prime numbers. Suppose that $f_j$ is a non-van holds. Here $\mathop{\mathrm{meas}}\nolimits$ denotes the Lebesgue measure on $\mathbb{R}$, and the

Figures (3)

  • Figure 1: The shift of the path of the integration
  • Figure 2: The rectangle $\mathcal{R}$
  • Figure 3: The shift of the path of integration

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 2.1
  • Corollary 2.2
  • Corollary 2.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 17 more