ON THE ZEROS OF THE RIEMANN ZETA-FUNCTION

Abstract

It is well known that the multiplicity of a complex zero ρ=β+iγ of the zeta-function is O(log[mid ]γ[mid ]). This may be proved by means of Jensen's formula, as in Titchmarsh [7, Chapter 9]. It may also be seen from the formula for the number N(T) of zeros such that 0<γ<T,

formula here

due to Backlund [1], in which E(T) is a continuous function satisfying E(T)=O(1/T) and

formula here

We assume here that T is not the ordinate of a zero; with appropriate definitions of N(T) and S(T) the formula is valid for all T. We have S(T)=O(logT). On the Lindelöf Hypothesis S(T)=o(logT), (Cramér [2]), and on the Riemann Hypothesis

formula here

(Littlewood [5]). These results are over 70 years old.

formula here

Because the multiplicity problem is hard, it seems worthwhile to see what can be said about the number of distinct zeros in a short T-interval. We obtain the following result, which is independent of any unproved hypothesis.