Zero-free regions near a line

TL;DR

This paper introduces metrics based on Hankel matrix positivity conditions to analyze how close entire functions of genus one are to being real rooted, with implications for the zeros of the Riemann zeta function.

Contribution

It develops explicit positivity-based metrics that relate zero spacing to the proximity of zeros to the real axis, providing new relaxations of the Riemann hypothesis.

Findings

Zeros must be far from the real axis under the conditions.

Relaxations of the Riemann hypothesis imply zeros are farther from the critical line.

Positivity conditions impose explicit zero spacing constraints.

Abstract

We analyze metrics for how close an entire function of genus one is to being real rooted. These metrics arise from truncated Hankel matrix positivity-type conditions built from power series coefficients at each real point. Specifically, if such a function satisfies our positivity conditions and has well-spaced zeros, we show that all of its zeros have to (in some explicitly quantified sense) be far away from the real axis. The obvious interesting example arises from the Riemann zeta function, where our positivity conditions yield a family of relaxations of the Riemann hypothesis. One might guess that as we tighten our relaxation, the zeros of the zeta function must be close to the critical line. We show that the opposite occurs: any potential complex zeros are forced to be farther and farther away from the critical line.