Limitations to mollifying $\zeta(s)$

TL;DR

This paper investigates the fundamental limits of mollifying the Riemann zeta-function on the critical line, providing bounds on off-diagonal contributions and linking mollified moments to pair correlation under RH.

Contribution

It establishes the first non-trivial lower bounds for off-diagonal terms in mollified moments of (s) and connects these bounds to Montgomery's Pair Correlation Function assuming RH.

Findings

Non-trivial lower bounds for off-diagonal contributions

Limitations on mollifier length for effective mollification

Connection between mollified moments and pair correlation under RH

Abstract

We establish limitations to how well one can mollify the Riemann zeta-function on the critical line with mollifiers of arbitrary length. Our result gives a non-trivial lower bound for the contribution of the off-diagonal terms to mollified moments of \zeta(s). On the Riemann Hypothesis, we establish a connection between the mollified moment and Montgomery's Pair Correlation Function.