EXPLICIT FORMULAE AND DISCREPANCY ESTIMATES FOR a-POINTS OF THE RIEMANN ZETA-FUNCTION

摘要

For a fixed a ≠ 0, an a-point of the Riemann zeta-function is a complex number ρ_a = β_a + iγ_a such that ζ(ρa)= a. Recently J. Steuding estimated the sum ∑0<γa≤T β_a>0 x~(ρa) for a fixed x as T →∞, and used this to prove that the ordinates γ_a are uniformly distributed modulo 1. We provide uniform estimates for this sum when x >0 and ≠1, and T >1. Using this, we bound the discrepancy of the sequence λγ_a when λ ≠ 0. We also find explicit representations and bounds for the Dirichlet coefficients of the series 1/(ζ(s)-a) and upper bounds for the abscissa of absolute convergence of this series.