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

TL;DR

The paper analyzes the analytic component of the error term in the summatory divisor function, extending the arithmetic–analytic decomposition used for the totient to $\sigma_1(n)$. It expresses the analytic part $E_{\sigma_1}^{AN}(x)$ via a Mellin transform tied to $\zeta(s)\zeta(s-1)$ and obtains an explicit RH-dependent bound $E_{\sigma_1}^{AN}(x) \ll x^{\delta'} \exp\left( \frac{\log x}{\log\log x} \right)$ with $\delta' = \max\{1/2, \delta\}$, as well as a uniform bound $E_{\sigma_1}^{AN}(x) \ll_{\varepsilon} x^{\delta'+\varepsilon}$. The proofs hinge on contour integration, RH-based bounds for zeta-functions on critical lines, and the inverse Mellin transform, paralleling prior work for $\varphi(n)$ while highlighting distinct features of the divisor-function case. These results contribute to RH-type characterizations of analytic components in divisor-sum errors and illuminate the limitations of such equivalences compared to the totient setting. Overall, the work advances precise control of analytic error terms in divisor-sum asymptotics and their connections to zeta-function theory.

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).

Theorems & Definitions (10)

• Theorem 1.1.1: Theorem 1.7 in Kac and Wie
• Theorem 1.2.1
• Lemma 1.2.2: Lemma 2.5 in K
• Theorem 1.2.3
• Theorem 2.1.1: Theorem in I
• Lemma 2.1.2: Lemma 1 in I
• Remark 2.1.1
• Lemma 3.1.1
• Lemma 3.1.2
• Remark 4.2.1