Statement of results. We assume the Riemann Hypothesis (RH) through out this paper; e=!+iy denotes a nontrivial zero of the Riemann zeta function. Our object is to investigate the distribution of the differences y -y' between the zeros. It would thus be desirable to know the Fourier transform ofthe distribution function of the numbers yy'; with this in mind we take

Prime numbers are the multiplicative building blocks of natural numbers. Understanding their overall influence and especially their distribution gives rise to central questions in mathematics and physics. In particular their finer distribution is closely connected with the Riemann hypothesis, the most important unsolved problem in the mathematical world. Assuming only subjects covered in a standard degree in mathematics, the authors comprehensively cover all the topics met in first courses on multiplicative number theory and the distribution of prime numbers. They bring their extensive and distinguished research expertise to bear in preparing the student for intelligent reading of the more advanced research literature. This 2006 text, which is based on courses taught successfully over many years at Michigan, Imperial College and Pennsylvania State, is enriched by comprehensive historical notes and references as well as over 500 exercises.

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( 1) We prove a corresponding result for theta functions. For real a > 0, let This function satisfies the functional equation(This may be proved by using the formula (4) below, and then twice applying the identity (8)