On some analytic properties of a function associated with the Selberg class satisfying certain special conditions

TL;DR

This paper investigates the analytic properties of a generalized function linked to the Selberg class, extending previous work on related functions and focusing on those with polynomial Euler products under specific conditions.

Contribution

It introduces a detailed analysis of the analytic behavior of a new class of functions associated with the Selberg class, expanding understanding of their properties.

Findings

Describes the analytic behavior of the generalized function $f(z,F)$ for Im$z > 0$

Extends previous results to functions with polynomial Euler products

Provides conditions under which the analytic properties hold

Abstract

In 2001, M.Rekos described the analytic behavior for a function $f(z)$ connected with the Euler totient function for Im$z > 0$ (see (1.2)) imitating the previous research of [1] and [3]. In the present paper, for Im$z > 0$ we describe the analytic behavior of the generalized function $f(z,F)$ (see (2.1)), where the function $F$ belongs to the subclass of the Selberg class which has a polynomial Euler product and satisfies some special conditions.