On the Density Hypothesis for the Selberg class

TL;DR

This paper proves a zero density theorem for the Selberg class, confirming the Density Hypothesis in certain cases and providing weaker results otherwise, advancing understanding of zeros of these functions.

Contribution

It establishes a general zero density theorem for the Selberg class, verifying the Density Hypothesis for functions of degree up to 2 in a specific strip.

Findings

Verifies the Density Hypothesis for degree ≤ 2 in the strip Re(s) ≥ 0.9

Provides weaker density results for higher degrees

Advances zero distribution understanding in the Selberg class

Abstract

We prove a general zero density theorem on the Selberg class of functions. The result verifies the Density Hypothesis in the strip when the real part of the variable is at least 0.9 under the assumption that the degree of the function does not exceed 2. In the other cases we obtain a density theorem weaker than the Density Hypothesis.