Zeta zero dependence and the critical line

TL;DR

This paper investigates the statistical dependence of the Riemann zeta function's magnitude around zeros, revealing a unique structure on the critical line and its absence off it, with implications for L-functions.

Contribution

It demonstrates the existence of a conditional distribution of the zeta function's magnitude on the critical line and its non-existence off it, using elementary probabilistic and number-theoretic methods.

Findings

Conditional distribution exists on the critical line.

No such distribution exists off the critical line.

Results extend to L-functions.

Abstract

On the critical line the conditional distribution of the zeta function's magnitude around zeta zeros exists and predicts the well-known pair correlation between nontrivial zeta zeros. However, this conditional distribution does not exist at most distances above or below any nontrivial zeta zeros that are off the critical line. This shows that the zeta function's magnitude cannot have vertical statistical structure at most distances around nontrivial zeta zeros off the critical line. The proofs of these results are straightforward, using only statistical properties of certain prime sums, elementary properties of normal and elliptical random variables, and the pole structure of the zeta function. These results readily generalize to L-functions.