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 $L$-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 $L$-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 $L$-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 $E$ 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 $E \rightarrow \infty$ 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

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 and Montgomery [‘Pair correlation of zeros and primes in short intervals’, Analytic number theory and Diophantine problems (Stillwater, 1984), Progress in Mathematics 70 (1987) 183–203] and Montgomery and Soundararajan [‘Primes in short intervals’, Comm. Math. Phys. 252 (2004) 589–617] 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 (for example, 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

Variance of sums in arithmetic progressions of arithmetic functions associated with higher degree <i>0</i>-functions in F_<i>q</i>[<i>t</i>]

We compute the variances of sums in arithmetic progressions of generalised -divisor functions related to certain -functions in q[], in the limit as q → ∞. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when q → ∞, in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual -divisor function, when the -function in question has degree one. They illustrate the role played by the degree of the -functions; in particular, we find qualitatively new behaviour when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over q[], and we illustrate them by examining in some detail the generalised -divisor functions associated with the Legendre curve