On the Number of Representations of Integers by various Quadratic and Higher Forms

TL;DR

This paper provides formulas and evaluations for the number of representations of integers by various quadratic and higher forms, including sums of cubes and quintic forms, and discusses generalized triangular numbers and the Gauss circle problem.

Contribution

It introduces new formulas and evaluations for representations of integers by quadratic, cubic, and quintic forms, and explores generalized triangular numbers and asymptotic behavior.

Findings

Formulas for representations by quadratic forms

Evaluations for sums of two cubes and quintic forms

Asymptotic formula for the Gauss circle problem

Abstract

We give formulas for the number of representations of non negative integers by various quadratic forms. We also give evaluations in the case of sum of two cubes (cubic case) and the quintic case, as well. We introduce a class of generalized triangular numbers and give several evaluations. Finally, we present a mean value asymptotic formula for the number of representations of an integer as sum of two squares known as the Gauss circle problem.