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

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 $0<\sigma \le 1$ 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 $0\le \sigma \le 1$.

37 pages; v7: version accepted by the journal

Cited by in corpus (3)

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