66 research outputs found
On the variance of sums of arithmetic functions over primes in short intervals and pair correlation for L-functions in the Selberg class
We establish the equivalence of conjectures concerning the pair correlation
of zeros of -functions in the Selberg class and the variances of sums of a
related class of arithmetic functions over primes in short intervals. This
extends the results of Goldston & Montgomery [7] and Montgomery & Soundararajan
[11] for the Riemann zeta-function to other -functions in the Selberg class.
Our approach is based on the statistics of the zeros because the analogue of
the Hardy-Littlewood conjecture for the auto-correlation of the arithmetic
functions we consider is not available in general. One of our main findings is
that the variances of sums of these arithmetic functions over primes in short
intervals have a different form when the degree of the associated -functions
is 2 or higher to that which holds when the degree is 1 (e.g. the Riemann
zeta-function). Specifically, when the degree is 2 or higher there are two
regimes in which the variances take qualitatively different forms, whilst in
the degree-1 case there is a single regime
Twin prime correlations from the pair correlation of Riemann zeros
We establish, via a formal/heuristic Fourier inversion calculation, that the
Hardy-Littlewood twin prime conjecture is equivalent to an asymptotic formula
for the two-point correlation function of Riemann zeros at a height on the
critical line. Previously it was known that the Hardy-Littlewood conjecture
implies the pair correlation formula, and we show that the reverse implication
also holds. A smooth form of the Hardy-Littlewood conjecture is obtained by
inverting the limit of the two-point correlation
function and the precise form of the conjecture is found by including
asymptotically lower order terms in the two-point correlation function formula.Comment: 11 page
The Selberg integral and a new pair-correlation function for the zeros of the Riemann zeta-function
The present paper is a report on joint work with Alessandro Languasco and
Alberto Perelli on our recent investigations on the Selberg integral and its
connections to Montgomery's pair-correlation function. We introduce a more
general form of the Selberg integral and connect it to a new pair-correlation
function, emphasising its relations to the distribution of prime numbers in
short intervals.Comment: Proceedings of the Third Italian Meeting in Number Theory, Pisa,
September 2015. To appear in the "Rivista di Matematica dell'Universita` di
Parma
Correlations of sieve weights and distributions of zeros
In this note we give two small results concerning the correlations of the
Selberg sieve weights. We then use these estimates to derive a new
(conditional) lower bound on the variance of the primes in short intervals, and
also on the so-called `form factor' for the pair correlations of the zeros of
the Riemann zeta function. Our bounds ultimately rely on the estimates of
Bettin--Chandee for trilinear Kloosterman fractions
From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond
The discovery of connections between the distribution of energy levels of
heavy nuclei and spacings between prime numbers has been one of the most
surprising and fruitful observations in the twentieth century. The connection
between the two areas was first observed through Montgomery's work on the pair
correlation of zeros of the Riemann zeta function. As its generalizations and
consequences have motivated much of the following work, and to this day remains
one of the most important outstanding conjectures in the field, it occupies a
central role in our discussion below. We describe some of the many techniques
and results from the past sixty years, especially the important roles played by
numerical and experimental investigations, that led to the discovery of the
connections and progress towards understanding the behaviors. In our survey of
these two areas, we describe the common mathematics that explains the
remarkable universality. We conclude with some thoughts on what might lie ahead
in the pair correlation of zeros of the zeta function, and other similar
quantities.Comment: Version 1.1, 50 pages, 6 figures. To appear in "Open Problems in
Mathematics", Editors John Nash and Michael Th. Rassias. arXiv admin note:
text overlap with arXiv:0909.491
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