Zeta zero dependence and the critical line

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.