A Fourier analysis of quadratic Riemann sums and Local integrals of \(\varvec{\zeta (s)}\)

Abstract

Let \(\zeta (s)\), \(s={\sigma }+it\), be the Riemann zeta function. We use Fourier analysis to obtain, after a preliminary study of quadratic Riemann sums, a precise formula of the local integrals \(\int _n^{n+1} |\zeta ({\sigma }+it ) |^2 \textrm{d}t\), for \(\frac{1}{2}<{\sigma }<1\). We also study related \(\mathcal {S}^{2}\)-Stepanov norms of \(\zeta (s)\) in connection with the strong Voronin Universality Theorem.