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Sums of Two Squares in Short Intervals

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Analytic Number Theory

Abstract

We show that there are short intervals [x, x + y] containing ≫ y 1∕10 numbers expressible as the sum of two squares, which is many more than the average when \(y = o((\log x)^{5/9})\). We obtain similar results for sums of two squares in short arithmetic progressions.

Dedicated to Helmut Maier on the occasion of his 60th birthday

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Notes

  1. 1.

    The Buchstab function ω(u) is defined by the delay-differential equation

    $$\displaystyle{\omega (u) = u^{-1}\quad (0 < u \leq 2),\qquad \qquad \frac{\partial } {\partial u}(u\omega (u)) =\omega (u - 1)\quad (u \geq 2).}$$
  2. 2.

    We ignore the elements of [n, n + h] which are a multiple of some prime p ≤ h, which is unavoidable.

  3. 3.

    Here a “small prime” refers to one bounded by roughly \(\exp (h^{-1+o(1)}\log x)\).

  4. 4.

    One needs to be slightly careful about the possible effect of Siegel zeros here, but this is a minor technical issue.

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Acknowledgements

The work in this paper was started whilst the author was a CRM-ISM Postdoctoral Fellow and the Université de Montréal, and was completed whilst he was a Fellow by Examination at Magdalen College, Oxford.

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Authors and Affiliations

  1. Magdalen College, Oxford, OX1 4AU, UK

    James Maynard

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  1. James Maynard
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Correspondence to James Maynard .

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Editors and Affiliations

  1. Department of Mathematics, Dartmouth College, Hanover, New Hampshire, USA

    Carl Pomerance

  2. Department of Mathematics, ETH-Zürich, Zürich, Switzerland

    Michael Th. Rassias

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Maynard, J. (2015). Sums of Two Squares in Short Intervals. In: Pomerance, C., Rassias, M. (eds) Analytic Number Theory. Springer, Cham. https://doi.org/10.1007/978-3-319-22240-0_15

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