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∣γ∣). 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, N ( T ) = T 2 π log T 2 π - T 2 π + S ( T ) + 7 8 + E ( T ) (1) due to Backlund [1], in which E(T) is a continuous function satisfying E(T)=O(1/T) and S ( T ) = 1 π arg ζ ( 1 2 + i T ) (2) 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 S ( T ) = O ( log T log log T ) (Littlewood [5]). These results are over 70 years old. 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.