Chapter 3: On the Differences Between Consecutive Prime Numbers, I

Abstract

We show by an inclusion-exclusion argument that the prime k-tuple conjecture of Hardy and Littlewood provides an asymptotic formula for the number of consecutive prime numbers which are a specified distance apart. This refines one aspect of a theorem of Gallagher that the prime k-tuple conjecture implies that the prime numbers are distributed in a Poisson distribution around their average spacing.

The pair correlation of zeros of DirichletL-functions and primes in arithmetic progressions

“…The modulus q is always supposed to be prime or 1. For q = 1 all of the results presented here yield the known estimates ( [2], [3], [4], [5], [7]) mentioned above. By the weaker form of the Conjecture it is meant that there is << in place of the asymptotic estimate in the Conjecture.…”

“…Assuming the Riemann Hypothesis Montgomery defined the pair correlation function of the critical zeros of ~(s) F(a,T) = (T log T)_I 2~ " ~ T'~(~-~')w(7 -7') (1) 0<*W~n<:T (;( 89 (where w(u) = 4 is a weighting function which serves to diminish the contribution from those pairs of zeros with large differences) and he proved that (see [7] and [2])…”

More Precise Pair Correlation of Zeros and Primes in Short Intervals

“…Assuming Riemann Hypothesis, he proved in [5] that, as T → ∞, F (X, T ) ∼ T 2π log X + T 2πX 2 (log T ) 2 for 1 ≤ X ≤ T (actually he only proved for 1 ≤ X ≤ o(T ) and the full range was done by Goldston [2]). He conjectured that F (X, T ) ∼ T 2π log T for T ≤ X which is known as the Strong Pair Correlation Conjecture.…”

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

“…for 1 ≤ x ≤ T (actually he only proved this for 1 ≤ x ≤ o(T ) and the full range was done by Goldston [5]). He conjectured that which draws connections with random matrix theory.…”