Further Estimates with Pseudogamma functions

TL;DR

This paper improves estimates of the doubly symmetric pseudo-Gamma function near the symmetry center, refining tools used in the proof of the density hypothesis for the Riemann zeta function.

Contribution

It provides sharper bounds for the pseudo-Gamma function on the real axis by a novel approach that reduces error in the density hypothesis proof.

Findings

Enhanced estimate accuracy for the pseudo-Gamma function near the symmetry center

Reduced error in the density hypothesis proof using a new approach

Improved understanding of the pseudo-Gamma function's symmetry properties

Abstract

The pseudo-Gamma function is a key tool introduced recently by Cheng and Albeverio in the proof of \break the density hypothesis. This function is doubly symmetric, which means that it is reflectively symmetric about the real axis by the Schwarz principle, whereas it is also reflectively symmmetric about the half line where the real part of the variable is equal to $\tfrac{1}{2}$. In this article, we sharpen the estimate given in the proof of the density hypothesis for this doubly symmetric pseudo-Gamma function on the real axis near the symmetry center by taking a different approach from the way used in the density hypothesis proof directly from the definition, reducing the error caused by the fact that the difference of two pivotal parameters in the definition of the pseudo-Gamma function is much larger than the difference of the variables in this particular case.