Unnormalized differences between zeros of L-functions

TL;DR

This paper investigates the distribution of differences between zeros of the Riemann zeta function and general L-functions, revealing how the zeros encode deep arithmetic information such as elliptic curve ranks.

Contribution

It introduces a measure explaining the distribution of zero differences and extends the analysis to zeros of general L-functions, linking zeros to arithmetic invariants.

Findings

Distribution of zero differences encodes zeros' locations

Elliptic curve rank is reflected in zeros of related L-functions

Established a measure connecting zeros' distribution to number-theoretic properties

Abstract

We study a subtle inequity in the distribution of unnormalized differences between imaginary parts of zeros of the Riemann zeta function. We establish a precise measure which explains the phenomenon, that the location of each Riemann zero is encoded in the distribution of large Riemann zeros. We also extend these results to zeros of more general L-functions. In particular, we show how the rank of an elliptic curve over Q is encoded in the sequences of zeros of other L-functions, not only the one associated to the curve.