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Riemann-Type Functional Equations -- Julia Line and Counting Formulae --

Athanasios Sourmelidis, Jörn Steuding, Ade Irma Suriajaya

Abstract

We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number $a\neq0$ and a function from the Selberg class $\mathcal{L}$, we prove a Riemann-von Mangoldt formula for the number of a-points of the $Δ$-factor of the functional equation of $\mathcal{L}$ and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these $a$-points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of $\mathcal{L}(s)$ taken at these points.

Riemann-Type Functional Equations -- Julia Line and Counting Formulae --

Abstract

We study Riemann-type functional equations with respect to value-distribution theory and derive implications for their solutions. In particular, for a fixed complex number and a function from the Selberg class , we prove a Riemann-von Mangoldt formula for the number of a-points of the -factor of the functional equation of and an analog of Landau's formula over these points. From the last formula we derive that the ordinates of these -points are uniformly distributed modulo one. Lastly, we show the existence of the mean-value of the values of taken at these points.

Paper Structure

This paper contains 1 section, 1 theorem, 10 equations.

Table of Contents

  1. Introduction and Statement of the Main Results

Key Result

Theorem 1

Let $a\neq0$ be a complex number, $\mathcal{L}\in\mathcal{S}^\sharp$ with $d_\mathcal{L}\geq1$ and $\psi(t):=\log t/\log\log t$, $t\geq3$. If $N_\pm(T;a,\Delta_\mathcal{L})$ counts the number of nontrivial $a$-points $\delta_a=\beta_a+i\gamma_a$ of $\Delta_{\mathcal{L}}(s)$ satisfying $0<\pm\gamma_a for any $T>0$.

Theorems & Definitions (1)

  • Theorem 1