Large gaps between sums of two squares

A. B. Kalmynin; S. V. Konyagin

Large gaps between sums of two squares

Number Theory 2022-04-27 v2

Authors: A. B. Kalmynin , S. V. Konyagin

Abstract

Let $\mathcal S=\{s_1<s_2<s_3<\ldots\}$ be the sequence of all natural numbers which can be represented as a sum of two squares of integers. For $X\ge2$ we denote by $g(X)$ the largest gap between consecutive elements of $\mathcal S$ that do not exceed $X$ . We prove that for $X \to +\infty$ the lower bound $g(X)\geq \left(\frac{390}{449}-o(1)\right)\ln X$ holds. This estimate is twice the recent estimate by R. Dietmann and C. Elsholtz.

Keywords

bound multiplicative functions collatz conjecture

Cite

@article{arxiv.1906.09100,
  title  = {Large gaps between sums of two squares},
  author = {A. B. Kalmynin and S. V. Konyagin},
  journal= {arXiv preprint arXiv:1906.09100},
  year   = {2022}
}

Comments

5 pages; This version will not be published in this form. It will appear in a joint and expanded manuscript by R. Dietmann, C. Elsholtz, A. Kalmynin, S. Konyagin and J. Maynard

Large gaps between sums of two squares

Abstract

Keywords

Cite

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