On the density of zeros of linear combinations of Euler products for $\sigma \gt {}$1

Abstract

It has been conjectured by Bombieri and Ghosh that the real parts of the zeros of a linear combination of two or more $L$-functions should be dense in the interval $\left[1,{\sigma }^{\ast }\right]$, where ${\sigma }^{\ast }$ 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 $\left[1,{\sigma }^{\ast }\right]$ whenever ${\sigma }^{\ast }>1$.