Assume the Riemann Hypothesis and let $F(X,T)=4\sum\limits_{0<\gamma_1,\gamma_2\leq T}\frac{X^{i(\gamma_1-\gamma_2)}}{4+(\gamma_1-\gamma_2)^2}$, where $\gamma_j$, $j=1,2$, run over the imaginary part of the non-trivial zeros of the Riemann zeta-function, be the Montgomery's pair correlation function. Goldston-Montgomery proved, for any $\epsilon>0$, that $ F(X,T) \sim \frac{1}{2\pi} T\log T$ uniformly for $X^\epsilon \leq T\leq X$ is equivalent to $J(X,H) \sim HX\log \frac{X}{H} \ \text{uniformly for} \ 1\leq H \leq X^{1-\epsilon}$ where $J(X,H)$ is Selberg's integral. Here we prove, for any $\epsilon>0$, that $ F(X,T) \sim \frac{1}{2\pi} T\log T$ uniformly for $X^{1/2+\epsilon}\leq T\leq X$ is equivalent to a suitable asymptotic formula for the truncated mean-square of exponential sums over primes.
Pair correlation of zeros, primes in short intervals and exponential sums over primes
LANGUASCO, ALESSANDRO;
2000
Abstract
Assume the Riemann Hypothesis and let $F(X,T)=4\sum\limits_{0<\gamma_1,\gamma_2\leq T}\frac{X^{i(\gamma_1-\gamma_2)}}{4+(\gamma_1-\gamma_2)^2}$, where $\gamma_j$, $j=1,2$, run over the imaginary part of the non-trivial zeros of the Riemann zeta-function, be the Montgomery's pair correlation function. Goldston-Montgomery proved, for any $\epsilon>0$, that $ F(X,T) \sim \frac{1}{2\pi} T\log T$ uniformly for $X^\epsilon \leq T\leq X$ is equivalent to $J(X,H) \sim HX\log \frac{X}{H} \ \text{uniformly for} \ 1\leq H \leq X^{1-\epsilon}$ where $J(X,H)$ is Selberg's integral. Here we prove, for any $\epsilon>0$, that $ F(X,T) \sim \frac{1}{2\pi} T\log T$ uniformly for $X^{1/2+\epsilon}\leq T\leq X$ is equivalent to a suitable asymptotic formula for the truncated mean-square of exponential sums over primes.File | Dimensione | Formato | |
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