Pair correlation and distribution of prime numbers.

Abstract

In early 1970s, Hugh Montgomery initialized the study of the pair correlation function <italic>F</italic>(<italic>X,T</italic>) on zeros of Riemann zeta function and made the famous Strong Pair Correlation Conjecture on its asymptotic size when <italic>T</italic> &le; <italic>X</italic>. Later, assuming Riemann Hypothesis, Goldston and Montgomery proved that this and two other asymptotic formulas for second moments of primes in short intervals are equivalent to each other. Recently, Montgomery and Soundararajan proposed a more precise asymptotic formula for one of the above second moments. They also worked on higher moments under Hardy-Littlewood Prime <italic>k</italic>-tuple Conjecture and came to the conclusion that primes in short intervals obey normal distribution. The first goal of this thesis is to make use of the more precise asymptotic formula described above to get more precise formulas for the other second moment and Strong Pair Correlation Conjecture. Our method is essentially that of Goldston & Montgomery but with explicit error terms. The second goal is to improve the asymptotic formulas for higher even moments which tells us how they deviate from normal distribution. The improved formulas match closely with actual data. The third goal is to get equivalence results on higher moments of primes in short intervals. For even moments, we can use the same method as that for second moments. However, difficulties arise for odd moments. The best one can do is to assume good information on even moments and use Ghosh's method. Despite many unproved assumptions, our results agree strongly with actual data. Finally, we try to get second order terms for <italic>F</italic>(<italic> X,T</italic>) in different ranges of <italic>X</italic>. When 1 &le; <italic> X</italic> &le; <italic>T</italic><super>1-&isin;</super>, we make more precise the original calculation by Montgomery. When <italic>T</italic><super> 1-&isin;</super> &le; <italic>X</italic>, we use Goldston & Gonek's method for computing mean values of Dirichlet polynomials and tails of Dirichlet series assuming Twin Prime Conjecture. From these, one gains better understanding on the behavior of <italic>F</italic>(<italic>X,T</italic>), especially when <italic> X</italic> &ap; <italic>T</italic>. Consequently, one has more precise formulations for the Strong and Weak Pair Correlation Conjectures which reinforce the belief that the zeros of the Riemann zeta function behave like the eigenvalues of a random matrix from the Gaussian Unitary Ensemble.