On certain sums over ordinates of zeta-zeros. II. (English) Zbl 1447.11092

Bondarenko, Andriy; Ivić, Aleksandar; Saksman, Eero; Seip, Kristian

Hardy-Ramanujan J. 41, 85-97 (2018).

Summary: Let \(\gamma\) denote the imaginary parts of complex zeros \(\rho=\beta+ i\gamma\) of \(\zeta(s)\). The problem of analytic continuation of the function \(G(s):=\sum_{\gamma >0} \gamma^{-s}\) to the left of the line \(\operatorname{Re} s = -1\) is investigated, and its Laurent expansion at the pole \(s=1\) is obtained. Estimates for the second moment on the critical line \(\int_1^T |G(\frac{1}{2}+ it)|^2 dt\) are revisited. This paper is a continuation of work begun by the second author [Bull., Cl. Sci. Math. Nat., Sci. Math. 122, No. 26, 39–52 (2001; Zbl 0999.11048)].

MSC:

11M06

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

Keywords:

Riemann zeta-function; Riemann hypothesis; analytic continuation; Laurent expansion; second moment

Citations:

Zbl 0999.11048