On a relation to the Riemann Hypothesis and an analytic part for the divisor function

TL;DR

This paper explores the connection between the Riemann Hypothesis and the divisor function's summatory behavior, providing explicit bounds under the hypothesis.

Contribution

It introduces an explicit bound for the analytic part of the divisor function's summatory function assuming the Riemann Hypothesis.

Findings

Established an upper bound of order x^{delta'} exp((log x)/(log log x))

Derived bounds under the assumption of the Riemann Hypothesis

Analyzed the relation between the Riemann Hypothesis and divisor function

Abstract

Let phi(n) denote the Euler totient function. We study the analytic part associated with the summatory function of sigma_1(n) and obtain explicit bounds under the Riemann Hypothesis. In particular, we establish an upper bound of order x^{delta'} exp((log x)/(log log x)), where delta' = max(1/2, delta).