Explicit $L^2$ bounds for the Riemann $\zeta$ function

TL;DR

This paper provides explicit $L^2$ bounds for the tails of the Riemann zeta function, using two different methods optimized for different ranges, aiding in rigorous integral computations involving $ eta$.

Contribution

It introduces two novel approaches for bounding weighted $L^2$ norms of $ eta$ tails, improving accuracy across different ranges of $T$ and $\sigma$.

Findings

Bounds are of the correct order for $0<\sigma extless 1$.

Methods are practical for rigorous computation of improper integrals.

Bounds for the $L^2$ norm of $ eta$ in $[1,T]$ are also provided.

Abstract

Explicit bounds on the tails of the zeta function $\zeta$ are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $\zeta$. Two approaches are followed, each giving the better result on a different range. The first one is inspired by the proof of the standard mean value theorem for Dirichlet polynomials. The second approach, superior for large $T$, is based on classical lines, starting with an approximation to $\zeta$ via Euler-Maclaurin. Both bounds give main terms of the correct order for $0<\sigma\leq 1$ and are strong enough to be of practical use for the rigorous computation of improper integrals. We also present bounds for the $L^{2}$ norm of $\zeta$ in $[1,T]$ for $0\leq\sigma\leq 1$.