On the density of zeros of linear combinations of Euler products for $\sigma>1$

TL;DR

This paper investigates the distribution of zeros of linear combinations of Euler products, disproving a conjecture about their density in certain intervals and describing their optimal configurations.

Contribution

It demonstrates that zeros' real parts are not always dense in the entire interval and characterizes the density in subintervals for orthogonal Euler products.

Findings

Zeros' real parts are not always dense in [1, σ*].

Density of zeros' real parts occurs in subintervals when σ*>1.

Provides the optimal configuration of zeros for orthogonal Euler products.

Abstract

It has been conjectured that the real parts of the zeros of a linear combination of two or more $L$-functions are dense in the interval $[1,\sigma^*]$, where $\sigma^*$ is the least upper bound of the real parts of such zeros. In this paper we show that this is not true in general. Moreover, we describe the optimal configuration of the zeros of linear combinations of orthogonal Euler products by showing that the real parts of such zeros are dense in subintervals of $[1,\sigma^*]$ whenever $\sigma^*>1$.