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Discrete universality for Matsumoto zeta-functions and the nontrivial zeros of the Riemann zeta-function

Keita Nakai

Abstract

In 2017, Garunkštis, Laurinčikas and Macaitienė proved the discrete universality theorem for the Riemann zeta-function sifted by the nontrivial zeros of the Riemann zeta-function. This discrete universality has been extended in various zeta-functions and $L$-functions. In this paper, we generalize this discrete universality for Matsumoto zeta-functions.

Discrete universality for Matsumoto zeta-functions and the nontrivial zeros of the Riemann zeta-function

Abstract

In 2017, Garunkštis, Laurinčikas and Macaitienė proved the discrete universality theorem for the Riemann zeta-function sifted by the nontrivial zeros of the Riemann zeta-function. This discrete universality has been extended in various zeta-functions and -functions. In this paper, we generalize this discrete universality for Matsumoto zeta-functions.
Paper Structure (4 sections, 10 theorems, 42 equations)

This paper contains 4 sections, 10 theorems, 42 equations.

Table of Contents

  1. Introduction and the statement of main results
  2. Preliminaries
  3. A limit theorem
  4. proof of main theorem

Key Result

Theorem 1.1

Let $\mathcal{K}$ be a compact set in the strip $1/2 < \sigma < 1$ with connected complement, and let $f(s)$ be a non-vanishing continuous function on $\mathcal{K}$ that is analytic in the interior of $\mathcal{K}$. Then, for any $\varepsilon > 0$ where $\mathrm{meas}$ denotes the 1-dimensional Lebesgue measure.

Theorems & Definitions (17)

  • Theorem 1.1: Voronin, Vo
  • Theorem 1.2: Garunkštis, Laurinčikas and Macaitienė GLM
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • ...and 7 more