Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function

TL;DR

This paper explores the relationship between zeros of the Riemann zeta function and its derivative, establishing conditions that influence class number bounds of imaginary quadratic fields.

Contribution

It provides a new link between zeros of the zeta function's derivative and the distribution of zeros of the zeta function itself, with implications for number theory.

Findings

Zeros of the derivative near the critical line imply many zeros of the zeta function.

Conditions on derivative zeros lead to lower bounds on class numbers.

The results connect zero distributions to algebraic number theory properties.

Abstract

We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields.