The prime number theorem and pair correlation of zeros of the Riemann zeta-function

Riemann ’ s zeta function and the prime number theorem

16 Riemann’s zeta function and the prime number theorem We now divert our attention from algebraic number theory to talk about zeta functions and L-functions. As we shall see, every global field has a zeta function that is intimately related to the distribution of its primes. We begin with the zeta function of the rational field Q, which we will use to prove the prime number theorem. We will ne...

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 conjecture. 1. Introduction. Assuming the Riemann Hypothesis (RH), let us denote by ˆ 1=2‡ ig a nontrivi...

Pair Correlation of the zeros of the Riemann zeta function in longer ranges

In this paper, we study a more general pair correlation function, F h (x, T), of the zeros of the Riemann zeta function. It provides information on the distribution of larger differences between the zeros.

On the Pair Correlation of Zeros of the Riemann Zeta-function

In 1972 Montgomery [20, 21] introduced a new method for studying the zeros of the Riemann zeta-function. One of his main accomplishments was to determine partially the pair correlation of zeros, and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros. Perhaps more importantly, he conjectured on number-theoretic grounds an asymptotic formula for the pai...

Prime Number Theory and the Riemann Zeta-Function