The average number of solutions of the Diophantine equation U^2 + V^2 = W^3 and related arithmetic functions

TL;DR

This paper derives precise asymptotic formulas with sharp error bounds for the summatory functions of specific arithmetic functions, including sums involving representations of integers as sums of squares and their cubes.

Contribution

It provides new asymptotic results with sharp error terms for the Dirichlet summatory functions of arithmetic functions related to sums of squares and their powers.

Findings

Asymptotic formulas with sharp error terms for r^2(n) and r(n^3) sums

Improved understanding of the distribution of representations of integers as sums of squares

Enhanced techniques for analyzing Dirichlet summatory functions

Abstract

This paper provides asymptotics with a sharp error term for the Dirichlet summatory function of a certain class of arithmetic functions. The result applies, e.g., to the sums over r^2(n) and r(n^3), where r(m) denotes the number of ways to write m as a sum of two squares of integers.