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Lemke Oliver and Soundararajan bias for consecutive sums of two squares

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Abstract

In a surprising recent work, Lemke Oliver and Soundararajan noticed how experimental data exhibits erratic distributions for consecutive pairs of primes in arithmetic progressions, and proposed a heuristic model based on the Hardy–Littlewood conjectures containing a large secondary term, which fits the data very well. In this paper, we study consecutive pairs of sums of squares in arithmetic progressions, and develop a similar heuristic model based on the Hardy–Littlewood conjecture for sums of squares, which also explains the biases in the experimental data. In the process, we prove several results related to averages of the Hardy–Littlewood constant in the context of sums of two squares.

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Notes

  1. In the case of square-free numbers, the Hardy–Littlewood conjecture is a theorem [26], and the analogue of [20] has been proved recently by Mennema [25].

References

  1. Aryan, F.: The distribution of k-tuples of reduced residues. Mathematika 61(1), 72–88 (2015)

    Article  MathSciNet  Google Scholar 

  2. Aryan, F.: Distribution of squares modulo a composite number. Int. Math. Res. Not. IMRN 23, 12405–12431 (2015)

    MathSciNet  MATH  Google Scholar 

  3. Balog, A., Wooley, T.D.: Sums of two squares in short intervals. Can. J. Math. 52(4), 673–694 (2000)

    Article  MathSciNet  Google Scholar 

  4. Bank, E., Bary-Soroker, L., Fehm, A.: Sums of two squares in short intervals in polynomial rings over finite fields. Am. J. Math. 140(4), 1113–1131 (2018)

    Article  MathSciNet  Google Scholar 

  5. Banks, W.D., Freiberg, T., Turnage-Butterbaugh, C.L.: Consecutive primes in tuples. Acta Arith. 167(3), 261–266 (2015)

    Article  MathSciNet  Google Scholar 

  6. Bary-Soroker, L., Fehm, A.: Correlations of sums of two squares and other arithmetic functions in function fields. Int. Math. Res. Not. 2019(14), 4469–4515 (2019)

    Article  MathSciNet  Google Scholar 

  7. Bary-Soroker, L., Smilansky, Y., Wolf, A.: On the function field analogue of Landau’s theorem on sums of squares. Finite Fields Appl 39, 195–215 (2016)

    Article  MathSciNet  Google Scholar 

  8. Cazaran, J., Moree, P.: On a claim of Ramanujan in his first letter to Hardy. Exposit. Math. 17(4), 289–311 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Ciolan, A., Languasco, A., Moree, P.: Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms. preprint (2021). https://arxiv.org/pdf/2109.03288.pdf

  10. Connors, R.D., Keating, J.P.: Two-point spectral correlations for the square billiard. J. Phys. A 30(6), 1817–1830 (1997)

    Article  MathSciNet  Google Scholar 

  11. Eriksson, I., Gustafsson, L.: On the asymptotic behaviour of sums of two squares, p. 31. KTH Royal Institute of Technology, School of Engineering Sciences, Degree project (2018)

  12. Freiberg, T., Kurlberg, P., Rosenzweig, L.: Poisson distribution for gaps between sums of two squares and level spacings for toral point scatterers. Commun. Number Theory Phys. 11(4), 837–877 (2017)

    Article  MathSciNet  Google Scholar 

  13. Gallagher, P.X.: On the distribution of primes in short intervals. Mathematika 23(1), 4–9 (1976)

    Article  MathSciNet  Google Scholar 

  14. Gorodetsky, O., Rodgers, B.: The variance of the number of sums of two squares in \(\mathbb{F}_q[T]\) in short intervals. J. AMS (to appear)

  15. Granville, A.: Least primes in arithmetic progressions. Théorie des nombres. In: Proceedings of the International Conference held at the Université Laval, Quebec, Quebec, July 5–18, 1987, Edited by Jean-Marie De Koninck and Claude Levesque. Walter de Gruyter & Co., Berlin, (1989), 306–321

  16. Hooley, C.: On the intervals between numbers that are sums of two squares. Acta Math. 127, 279–297 (1971)

    Article  MathSciNet  Google Scholar 

  17. Hooley, C.: On the intervals between numbers that are sums of two squares. II. J. Number Theory 5, 215–217 (1973)

    Article  MathSciNet  Google Scholar 

  18. Koukoulopoulos, D.: The distribution of prime numbers. Graduate Studies in Mathematics, 203. American Mathematical Society, Providence, xii + 356 (2019)

  19. Landau, E.: Über die einteilung der positiven ganzen zahlen in vier klassen nach der mindestzahl der zu ihrer additiven zusammensetzung erforderlichen quadrate. Arch. Math u. Phys. (3) 13, 305–312 (1908)

    MATH  Google Scholar 

  20. Lemke Oliver, R.J., Soundararajan, K.: Unexpected biases in the distribution of consecutive primes. Proc. Natl. Acad. Sci. USA 113, no. 31, E4446–E4454 (2016)

  21. Lucia (https://mathoverflow.net/users/38624/lucia), Asymptotic density of k-almost primes, URL (version: 2013-10-31). https://mathoverflow.net/q/146469

  22. Maier, H.: Primes in short intervals. Mich. Math. J. 32(2), 221–225 (1985)

    Article  MathSciNet  Google Scholar 

  23. Mathematica, Version 12.0, Wolfram Research Inc. (2020). https://www.wolfram.com/mathematica/

  24. Maynard, J.: Dense clusters of primes in subsets. Compos. Math. 152(7), 1517–1554 (2016)

    Article  MathSciNet  Google Scholar 

  25. Mennema, I.: The Distribution of Consecutive Square-Free Numbers, Master Thesis, Department of Mathematics Leiden University, pp 31 (2017)

  26. Mirsky, L.: On the frequency of pairs of square-free numbers with a given difference. Bull. Am. Math. Soc. 55, 936–939 (1949)

    Article  MathSciNet  Google Scholar 

  27. Montgomery, H.L., Vaughan, R.C.: A basic inequality. In: Proceedings of the Congress on Number Theory (Spanish) (Zarauz, 1984), 163–175, Univ. País Vasco-Euskal Herriko Unib., Bilbao (1989)

  28. Montgomery, H.L., Soundararajan, K.: Primes in short intervals. Comm. Math. Phys. 252(1–3), 589–617 (2004)

    Article  MathSciNet  Google Scholar 

  29. Montgomery, H.L., Vaughan, R.C.: On the distribution of reduced residues. Ann. Math. (2) 123(2), 311–333 (1986)

    Article  MathSciNet  Google Scholar 

  30. Ramachandra, K.: Some problems of analytic number theory. Acta Arithmetica 31, 313–324 (1976)

    Article  MathSciNet  Google Scholar 

  31. Rieger, G.J.: Über die Anzahl der als Summe von zwei Quadraten darstellbaren und in einer primen Restklasse gelegenen Zahlen unterhalb einer positiven Schranke. II. (German). J. Reine Angew. Math. 217, 200–216 (1965)

    Article  MathSciNet  Google Scholar 

  32. SageMath, the Sage Mathematics Software System (Version 8.6.rc0), The Sage Developers (2019). https://www.sagemath.org

  33. Shanks, D.: The second-order term in the asymptotic expansion of \({B}(x)\). Math. Comp. 18, 75–86 (1964)

    MathSciNet  MATH  Google Scholar 

  34. Shiu, D.K.L.: Strings of congruent primes. J. Lond. Math. Soc. (2) 61(2), 359–373 (2000)

    Article  MathSciNet  Google Scholar 

  35. Smilansky, Y.: Sums of two squares—pair correlation and distribution in short intervals. Int. J. Number Theory 9(7), 1687–1711 (2013)

    Article  MathSciNet  Google Scholar 

  36. Stanley, G.K.: Two assertions made by Ramanujan. J. Lond. Math. Soc. 3(3), 232–237 (1928)

    Article  MathSciNet  Google Scholar 

  37. Stanley, G.K.: Corrigenda: two assertions made by Ramanujan. J. Lond. Math. Soc. 4(1), 32 (1929)

    Article  MathSciNet  Google Scholar 

  38. Tenenbaum, G.: Introduction to analytic and probabilistic number theory. Third edition. Translated from the 2008 French edition by Patrick D. F. Ion. Graduate Studies in Mathematics, 163. American Mathematical Society, Providence, RI, xxiv+629 pp (2015)

  39. The PARI Group, PARI/GP version 2.11.1, Univ. Bordeaux (2019). http://pari.math.u-bordeaux.fr/

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Acknowledgements

We thank the organizers of the MOBIUS ANT who invited us to present a preliminary version of the results contained in this paper in a friendly environment with subsequent discussions that led to several improvements in our exposition. We are particularly grateful to Dimitris Koukoulopoulos for constructive discussions, and for pointing out some relevant material from his book [18], and the post of Lucia on Mathoverflow [21]. We thank the anonymous referee for helpful comments.

Our results and conjectures are supported by several numerical data, that were computed using SageMath [32], PARI/GP [39] and Mathematica [23]. The authors thank the Centre de Recherches Mathématiques (CRM) in the Université de Montréal for offering their clusters for some of the numerical computations.

The first author was supported by a NSERC Discovery Grant and a FQRNT Team Grant; the second author was supported by the grant KAW 2019.0517 from the Knut and Alice Wallenberg Foundation for a post-doctoral position in Chalmers University of Technology and the University of Gothenburg; the fourth author was supported by a Concordia CUSRA.

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  1. Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Blvd. West, Montreal, Quebec, H3G 1M8, Canada

    Chantal David, Jungbae Nam & Jeremy Schlitt

  2. Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, Université Littoral Côte d’Opale, UR 2597 LMPA, 62100, Calais, France

    Lucile Devin

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  1. Chantal David
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Correspondence to Lucile Devin.

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Communicated by Roseline Periyanayagam.

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David, C., Devin, L., Nam, J. et al. Lemke Oliver and Soundararajan bias for consecutive sums of two squares. Math. Ann. 384, 1181–1242 (2022). https://doi.org/10.1007/s00208-021-02307-2

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