Bounding zeta on the 1-line under the partial Riemann hypothesis

TL;DR

This paper establishes explicit bounds for the Riemann zeta-function along the line Re(s)=1, assuming the Riemann hypothesis holds up to a certain height, thereby refining existing bounds in finite regions.

Contribution

It provides improved explicit bounds for the zeta-function and its reciprocal on the line Re(s)=1 under a partial Riemann hypothesis assumption.

Findings

Enhanced bounds for the logarithmic derivative of zeta

Refined bounds for the reciprocal of zeta in finite regions

Results depend on the height T up to which RH is assumed true

Abstract

We provide explicit bounds in the theory of the Riemann zeta-function at the line $\Re{s}=1$, assuming that the Riemann hypothesis holds until the height $T$. In particular, we improve some bounds, in finite regions, for the logarithmic derivative and the reciprocal of the Riemann zeta-function.