Explicit $L^2$ bounds for the Riemann $ζ$ function

Abstract

Explicit bounds on the tails of the zeta function $ζ$ are needed for applications, notably for integrals involving $ζ$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $ζ$. 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 $ζ$ via Euler-Maclaurin. Both bounds give main terms of the correct order for $0<σ\leq 1$ 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 $ζ$ in $[1,T]$ for $0\leqσ\leq 1$.