The pair correlation of zeros of the Riemann zeta function and distribution of primes

Assuming a special version of the Montgomery-Odlyzko law on the pair correlation of zeros of the Riemann zeta function conjectured by Rudnick and Sarnak and assuming the Riemann Hypothesis, we prove new results on the prime number theorem, difference of consecutive primes, and the twin prime con- jecture. In this article, we will show that a version of the above conjecture for the pair correlation of zeros of the zeta function zÖsÜ implies interesting arithmetical results on prime distribution (Theorems 2, 3, and 4). These results can give us deep insight on possible ultimate bounds of these prime distribution problems. One can also see that the pair (and n- level) correlation of zeros of zeta and L-functions is a powerful method in number theory. Our computation shows that the test function f and the support of its Fourier transform b play a crucial role in the conjecture. To see the conjecture in Rudnick and Sarnak (13) in the case of the zeta function zÖsÜ and nà 2, the pair correlation, we use a test function fÖx; yÜ which satisfies the following three conditions: