## Primes in almost all short intervals and the distribution of the zeros of the Riemann zeta-function

Citations: | 1 - 1 self |

### BibTeX

@MISC{Zaccagnini_primesin,

author = {Alessandro Zaccagnini},

title = {Primes in almost all short intervals and the distribution of the zeros of the Riemann zeta-function},

year = {}

}

### OpenURL

### Abstract

We study the relations between the distribution of the zeros of the Riemann zeta-function and the distribution of primes in "almost all" short intervals. It is well known that a relation like #(x)-#(x-y) holds for almost all x [N, 2N ] in a range for y that depends on the width of the available zero-free regions for the Riemann zeta-function, and also on the strength of density bounds for the zeros themselves. We also study implications in the opposite direction: assuming that an asymptotic formula like the above is valid for almost all x in a given range of values for y, we find zero-free regions or density bounds.