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On certain sums over ordinates of zeta-zeros. (English) Zbl 0999.11048

For \(\operatorname{Re} s>1\) let \(G(s)\) be the function defined by \(G(s)=\sum_{\gamma>0}\gamma^{-s}\), where \(\gamma\) denotes the ordinates of complex zeros of the Riemann zeta-function \(\zeta(s)\), and for \(0<\operatorname{Re} s\leq 1\) by analytic continuation. The main result is: For \(\sigma\) fixed we have \[ \int_1^T|G(\sigma+it)|^2 dt\ll\begin{cases} T,&1/2<\sigma\leq 1 \\T\log^2T, &\sigma=1/2\\T^{2-2\sigma}\log T,&0<\sigma<1/2.\end{cases} \] Certain integrals involving the function \(S(T)\), that is closely related to \(G(s)\), are also considered.
For Part II, see [A. Bondarenko et al., Hardy-Ramanujan J. 41, 85–97 (2018; Zbl 1447.11092)].

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)

Citations:

Zbl 1447.11092
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