Maximal order for divisor functions and zeros of the Riemann zeta-function

TL;DR

This paper explores the connection between the maximal order of divisor functions and the zeros of the Riemann zeta-function, providing new equivalent conditions for the Riemann hypothesis based on divisor sums.

Contribution

It establishes a novel link between zero-free regions of the zeta-function and the maximal order of the sum of powers of divisors, offering new criteria for the Riemann hypothesis.

Findings

Equivalent conditions for the Riemann hypothesis involving the sum of 1/2-th powers of divisors

Relations between zero-free regions of the zeta-function and divisor function maximal order

Conditions involving partial Euler products for the Riemann zeta-function

Abstract

Ramanujan investigated maximal order for the number of divisors function by introducing some notion such as (superior) highly composite numbers. He also studied maximal order for other arithmetic functions including the sum of powers of divisors function. In this paper we relate zero-free regions for the Riemann zeta-function to maximal order for the sum of powers of divisors function. In particular, we give equivalent conditions for the Riemann hypothesis in terms of the sum of the $1/2$-th powers of divisors function. As a by-product, we also give equivalent conditions for the Riemann hypothesis in terms of the partial Euler product for the Riemann zeta-function.