More Precise Pair Correlation of Zeros and Primes in Short Intervals

Abstract

Goldston and Montgomery [3] proved that the Strong Pair Correlation Conjecture and two second moments of primes in short intervals are equivalent to each other under Riemann Hypothesis. In this paper, we get the second main terms for each of the above and show that they are almost equivalent to each other.

On the variance of sums of arithmetic functions over primes in short intervals and pair correlation forL-functions in the Selberg class

“…The equivalence between Conjecture 1.1 and Conjecture 1.2 has been investigated further in [3,9] to include the lower order terms.…”

The Selberg integral and a new pair-correlation function for the zeros of the Riemann zeta-function

“…Notice that for θ = 1/x this strong result is essentially empty! Notice also what happens for θ = 1: the average bound is slightly stronger than the pointwise one given in (2).…”

“…• Goldston & Montgomery (1987) [6] • Montgomery & Soundararajan (2002) [16] • Chan (2003) [2] • Languasco, Perelli & Z. (2012) [10].…”

“…We stress again the importance of the uniformity ranges in the hypothesis. Following Chan's results in [2], Languasco, Perelli and the author studied relations between hypothetical bounds of the type…”

Explicit relations between primes in short intervals and exponential sums over primes

“…We remark that the uniformity ranges here are smaller than the ones in [2] and that it is due to the presence of E(X, h) (a term which naturally comes from Gallagher's lemma), see (7) and Lemma 3. In 2003 Chan [1] formulated a more precise pair-correlation hypothesis and gave explicit results for the connections between the error terms in the asymptotic formulae for F (X, T ) and J(X, h). Such results were recently extended and improved by the authors of this paper in a joint work with Perelli [7]: writing…”