Zeta zeros and Dirichlet L-functions
The author continues his earlier work on the uniformity of distribution of zeros of the Riemann zeta-function and Dirichlet L-functions [Comment. Math. Univ. St. Pauli 31, 99-113 (1982; Zbl 0489.10032) and Proc. Japan Acad., Ser. A 63, 370-373 (1987; Zbl 0636.10032)]. He assumes the Riemann hypothesis and proves three theorems. The first evaluates a certain sum of values of Dirichlet L-functions over imaginary parts of zeros of \(\zeta\) (s). The second theorem specializes the result of the first to \(\rho\) (s). It states that, for any fixed integer \(K\geq 2\), \[ \lim_{T\to \infty}2\pi T^{-1}\sum_{0<\gamma \leq T}(\zeta (\frac{1}{2}+\frac{i\gamma}{K})-1)=\frac{\zeta '}{\zeta}(\frac{K+1}{2}). \] The third theorem gives a relation equivalent to the Generalized Riemann Hypothesis for all Dirichlet functions L(s,\(\chi)\).
scientific article; zbMATH DE number 4083744