# Higher Correlations and the Alternative Hypothesis

## TL;DR

This paper investigates whether higher correlation functions of zeta zeros can disprove the Alternative Hypothesis, constructing a counterexample point process that mimics known statistics but satisfies the hypothesis.

## Contribution

It constructs an explicit counterexample point process showing higher correlations cannot rule out the Alternative Hypothesis, complementing recent independent results by Tao.

## Key findings

- Higher correlation functions are insufficient to disprove the Alternative Hypothesis.
- A counterexample point process mimics known zeta zero statistics while satisfying the hypothesis.
- Existence of a deterministic set of points in half-integers satisfying the hypothesis.

## Abstract

The Alternative Hypothesis concerns a hypothetical and unlikely picture of how zeros of the Riemann zeta function are spaced which one would like to rule out. In the Alternative Hypothesis, the renormalized distance between nontrivial zeros is supposed to always lie at a half integer. It is known that the Alternative Hypothesis is compatible with what is known about the pair correlation function of zeta zeros. We ask whether what is currently known about higher correlation functions of the zeros is sufficient to rule out the Alternative Hypothesis and show by construction of an explicit counterexample point process that it is not. A similar result was recently independently obtained by T. Tao, using slightly different methods. We also apply the ergodic theorem to this point process to show there exists a deterministic collection of points lying in $\tfrac{1}{2}\mathbb{Z}$ which satisfy the Alternative Hypothesis spacing but mimic all statistics which are currently known about zeros of the zeta function.