On the gaps between the consecutive zeros of the Riemann zeta function
Assuming the Riemann Hypothesis it is shown that \[ \sum_{\gamma_ n\leq T}(\gamma_{n+1}-\gamma_ n)^ 2\leq (9+o(1))T^ 2N(T)^{-1}, \] in the usual notation. Such an estimate, with an unspecified constant, is known to hold unconditionally [see the author, Bull. Am. Math. Soc. 81, 139-142 (1975; Zbl 0297.10026), for example]. The proof depends on a formula for \[ \int^{T}_{0}(S(t+\Delta)-S(t))^ 2dt, \] where \(\Delta =o(1)\), in which there is a main term involving Montgomery's pair correlation function F(a,T), and an error term o(1). The proof of this depends on work of \textit{D. A. Goldston} [J. Number Theory 27, 149-177 (1987; Zbl 0618.10037)]. To bound the main term one uses the estimate \[ \int^{\infty}_{1}F(a,T)a^{-2}da<2 \] for \(T\geq T_ 0\), also due to Goldston (loc. cit.).
scientific article; zbMATH DE number 4173112
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