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

Let \(\zeta (s)\) $\zeta (s)$ , \(s={\sigma }+it\) $s=\sigma +it$ , 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 \(\int _n^{n+1} |\zeta ({\sigma }+it ) |^2 \textrm{d}t\) ${\int }_n^{n+1}{|\zeta (\sigma +it)|}^2\text{d}t$ , for \(\frac{1}{2}<{\sigma }<1\) $\frac{1}{2}<\sigma <1$ . We also study related \(\mathcal {S}^{2}\) ${\mathcal{S}}^2$ -Stepanov norms of \(\zeta (s)\) $\zeta (s)$ in connection with the strong Voronin Universality Theorem.