Prime zeta function statistics and Riemann zero-difference repulsion
From MaRDI portal
Publication:6359769
DOI10.1088/1742-5468/AC0EE0arXiv2102.02280WikidataQ114096200 ScholiaQ114096200MaRDI QIDQ6359769
Publication date: 3 February 2021
Abstract: We present a derivation of the numerical phenomenon that differences between the Riemann zeta function's nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical "repulsion" between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarithm of the Riemann zeta function's magnitude on the 1-line. This creates notable negative covariance at separations approximately equal to the imaginary parts of the Riemann zeros. This covariance function and the singularity structure of the prime zeta function combine to create a conditional statistical bias at the locations of the Riemann zeros that predicts the zero-difference repulsion effect. Our method readily generalizes to describe similar effects in the zeros of related L-functions.
This page was built for publication: Prime zeta function statistics and Riemann zero-difference repulsion
Report a bug (only for logged in users!)Click here to report a bug for this page (MaRDI item Q6359769)
