© 1999 by London Mathematical Society
© The London Mathematical Society
On the Zeros of the Riemann Zeta-Function
Department of Mathematics, University of York York YO1 5DD
Received 11 October 1996. Revision received 11 April 1997. Revision received 31 October 1997.
It is well known that the multiplicity of a complex zero 
=ß+i
of the zeta-function is O(log||). 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<<T,
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due to Backlund [1], in which E(T) is a continuous function satisfying E(T)=O(1/T) and
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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
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(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.