Analytic expressions pertaining to certain arithmetical functions

TL;DR

This paper develops a method to derive analytic expressions for certain arithmetical sums related to solutions of specific Diophantine equations, with potential implications for the Riemann hypothesis.

Contribution

It introduces a general approach for obtaining explicit formulas for arithmetical sums over solutions of particular Diophantine equations, improving understanding of these sums.

Findings

Derived analytic expressions for sums over solutions of $kb^2+da^2=N$ and $kb^2-da^2=N$.

Proposed a possible enhancement to the Robin-Lagarias criteria for the Riemann hypothesis.

Established a framework connecting arithmetical sums and Diophantine equations.

Abstract

We demonstrate the general outlines of a method for obtaining analytic expressions for certain types of general arithmetical sums. In particular, analytical expressions for a general arithmetical sum whose terms are summed over either the positive integer solutions $(a,b)$ of the Diophantine equation $kb^2+da^2 = N$ or the positive integer solutions $(a,b)$ of the Diophantine equation $kb^2-da^2 = N$ are derived. As one of the consequences, we propose a possible improvement of the Robin-Lagarias criteria for the Riemann hypothesis.