Counting functions for sums of rational powers of integers

TL;DR

This paper develops counting functions for sums of rational powers of integers, providing a framework to analyze their distribution and deriving estimates, especially for sums of square roots, with connections to the Riemann zeta function.

Contribution

It introduces a novel method using convolution exponentials to count sums of rational powers, linking to the Riemann zeta function and deriving new estimates.

Findings

Constructed counting functions for sums of rational powers.

Established estimates for sums of square roots.

Connected counting functions to the Riemann zeta function.

Abstract

Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to a given $w>0$ is determined. The counting functions developed are framed in terms of convolution exponentials, and are closely related to the Riemann zeta function. At the conclusion, several estimates are derived, with special emphasis on the case of sums of square roots, i.e. $j=1$, $k=2$.