Real zeros of the Hurwitz zeta function

TL;DR

This paper extends the understanding of real zeros of the Hurwitz zeta function, generalizing previous results to all negative real numbers, beyond the known intervals.

Contribution

It generalizes Nakamura's existence condition for real zeros of the Hurwitz zeta function to all negative real numbers.

Findings

Identifies conditions for the existence of real zeros across all negative reals

Extends previous results from specific intervals to the entire negative real line

Provides a comprehensive framework for understanding zeros of the Hurwitz zeta function

Abstract

It is well known that real zeros of the Riemann zeta function are negative even integers. As for real zeros of the Hurwitz zeta function, T. Nakamura recently gave an existence condition in the intervals (0,1) and (-1,0). We generalize this result for all negative real numbers.