Counting functions for sums of rational powers of integers

Trevor Wine

Counting functions for sums of rational powers of integers

Number Theory 2018-12-21 v2

Authors: Trevor Wine

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. $j=1$ , $k=2$ .

Keywords

counting functions sums of rational powers integers convolution exponentials riemann zeta function square roots

Cite

@article{arxiv.1810.03038,
  title  = {Counting functions for sums of rational powers of integers},
  author = {Trevor Wine},
  journal= {arXiv preprint arXiv:1810.03038},
  year   = {2018}
}

Comments

26 pages, 6 figures, PDFLaTeX; minor improvements to exposition; corrected typos and a few relatively minor errors (almost all the corrections and improvements occurred in section 4)

Counting functions for sums of rational powers of integers

Abstract

Keywords

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