## 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 $$L(s, \chi)$$

Zbl 1447.11092
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