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

TL;DR

This paper investigates Riemann-type functional equations within the Selberg class, deriving formulas for counting solutions and analyzing their distribution, including uniform distribution modulo one and mean-value properties.

Contribution

It introduces new counting and distribution formulas for solutions of Riemann-type equations in the Selberg class, extending classical results to these generalized functions.

Findings

Derived a Riemann-von Mangoldt formula for a-points of the $ riangle$-factor.

Established an analog of Landau's formula for these points.

Proved the uniform distribution of the ordinates of a-points modulo one.

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 $\Delta$-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.