Fan Ge, Department of Pure Mathematics, University of Waterloo
The distribution of zeros of the derivative of the Riemann zeta-function is closely related to that of zeta itself. One of the basic questions in the study of zeros is the zero-counting problem. In particular, the error terms in the zero-counting formulas are of special interest. For the Riemann zeta-function the best known bound for the error term is O(log T) due to von Mangoldt in 1905. If we assume the Riemann Hypothesis (RH), then the essentially best bound is O(log T/loglog T) due to Littlewood in 1924. For the derivative of the Riemann zeta-function Berndt proved the bound O(log T) unconditionally in 1970, and assuming RH Akatsuka proved O(log T/sqrt(loglog T)) in 2012. We show that on RH, the error term in the zero-counting formula for the derivative of Riemann zeta is O(log T/loglog T), thus it is of the same size as that for zeta itself.
MC 5501