Some observations concerning the distribution of the zeros of the zeta functions. III
This is is the continuation of the author's extensive work on the distribution of zeros of the Riemann zeta-function \(\zeta(s)\) and Dirichlet \(L\)-functions \(L(s,\chi)\) [Part II, Comment. Math. Univ. St. Pauli 40, 125-231 (1991; Zbl 0743.11043)]. As before the author assumes the Generalized Riemann Hypothesis, and obtains a precise asymptotic formula for the sum \[ \sum_{Y < \gamma(\psi) \leq T}e^{ib\gamma(\psi)\log(b\gamma(\psi)/2\pi e\alpha)}, \] which is too complicated to be reproduced here. It is assumed that \(0 < \alpha \ll T\), \(1 \ll Y \leq T\), \(b > 0\), and \(\gamma(\psi)\) denotes imaginary parts of zeros of \(L(s,\psi)\) wher \(\psi\) is a primitive character \(\mod k\). If \(k = 1\), then \(L(s,\psi) = \zeta(s)\), and the author states two corollaries for \(\zeta(s)\) under the Riemann hypothesis.
exponential sums over zeros of zeta functions
scientific article; zbMATH DE number 97940