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Prime lattice points in ovals

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Abstract

We study the distribution of lattice points with prime coordinates lying in the dilate of a convex planar domain having smooth boundary, with nowhere vanishing curvature. Counting lattice points weighted by a von Mangoldt function gives an asymptotic formula, with the main term being the area of the dilated domain, and our goal is to study the remainder term. Assuming the Riemann Hypothesis, we give a sharp upper bound, and further assuming that the positive imaginary parts of the zeros of the Riemann zeta function are linearly independent over the rationals allows us to give a formula for the value distribution function of the properly normalized remainder term.

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Notes

  1. Smoothness breaks down if we only take a finite sum.

  2. There is an unfortunate typo in [1, equation (25)].

References

  1. Bennett, W.R.: Distribution of the sum of randomly phased components. Q. J. Appl. Math. 5, 385–393 (1948)

    Article  MathSciNet  Google Scholar 

  2. Best, D.G., Trudgian, T.S.: Linear relations of zeroes of the zeta-function. Math. Comp. 84(294), 2047–2058 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bleher, P.M.: On the distribution of the number of lattice points inside a family of convex ovals. Duke Math. J. 67, 461–481 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bleher, P.M., Dyson, F.J., Lebowitz, J.L.: Non-Gaussian energy level statistics for some integrable systems. Phys. Rev. Lett. 71(19), 3047–3050 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bleher, P.M.: Distribution of the error term in the Weyl asymptotics for the Laplace operator on a two-dimensional torus and related lattice problems. Duke Math. J. 70(3), 655–682 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Colin de Verdière, Y.: Nombre de points entiers dans une famille homothétique de domains de \({\bf R}\). Ann. Sci. Ecole Norm. Sup. 10(4), 559–575 (1977)

  7. Davenport, H.: Multiplicative number theory. In: Montgomery, H.L. (eds.) Revised and with a Preface. Graduate Texts in Mathematics, 3rd edn, vol. 74. Springer, New York, xiv+177 (2000)

  8. Flower, J.O.: Probability distributions of randomly-phased combinations of sine waves of equal amplitude. Appl. Acoust. 19(2), 131–138 (1986)

    Article  Google Scholar 

  9. Fujii, A.: An additive problem of prime numbers. II. Proc. Jpn. Acad. Ser. A Math. Sci 67(7), 248–252 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fujii, A.: An additive problem of prime numbers. III. Proc. Jpn. Acad. Ser. A Math. Sci 67(8), 278–283 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hlawka, E.: Über die Gleichverteilung gewisser Folgen, welche mit den Nullstellen der Zetafunktionen zusammenhängen. Sitzungsber. Österr. Akad. Wiss., Math.-Naturw. Kl. Abt. II 184, 459–471 (1975)

    MATH  Google Scholar 

  12. Heath-Brown, D.R.: The distribution and moments of the error term in the Dirichlet divisor problem. Acta Arith. 60(4), 389–415 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  13. Huxley, M.N.: Exponential sums and lattice points. III. Proc. Lond. Math. Soc. (3) 87(3), 591–609 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Littlewood, J.E.: Sur la distribution des nombres premiers. C. R. Acad. Sci. Paris 158, 1869–1872 (1914)

    MATH  Google Scholar 

  15. Nowak, W.G.: An \(\Omega \)-estimate for the lattice rest of a convex planar domain. Proc. R. Soc. Edinb. Sect. A 100(3–4), 295–299 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  16. Rademacher, H.A.: Fourier analysis in number theory, symposium on harmonic analysis and related integral transforms (Cornell University, Ithaca, N.Y., 1956). In: Rademacher, H. (ed.) Collected Papers of Hans Rademacher, vol. 2, pp. 434–458. Massachusetts Institute of Technology, Cambridge (1974)

    Google Scholar 

  17. Randol, B.: A lattice-point problem. Trans. Am. Math. Soc. 121, 257–268 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rayleigh, L.: On the problem of random vibrations, and of random flights in one, two or three dimensions. Philos. Mag. (6) 37, 321–347 (1919)

    Article  MATH  Google Scholar 

  19. Rubinstein, M.O., Sarnak, P.: Chebyshev’s bias. Exp. Math. 3(3), 173–197 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  20. Slack, M.: The probability distributions of sinusoidal oscillations combined in random phase. J. Inst. Elect. Eng. Part 3 93, 76–86 (1946)

    Google Scholar 

  21. Soundararajan, K.: Omega results for the divisor and circle problems. Int. Math. Res. Not. 2003(36), 987–1998 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Tsang, K.M.: Higher-power moments of \(\Delta (x)\), \(E(t)\) and \(P(x)\). Proc. Lond. Math. Soc. (3) 65(1), 65–84 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  23. van der Corput, J.G.: Neue zahlentheoretische Abschätzungen. Math. Ann. 89(3–4), 215–254 (1923)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wintner, A.: Upon a statistical method in the theory of Diophantine approximations. Am. J. Math. 55(1), 309–331 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  25. Wintner, A.: On the lattice problem of Gauss. Am. J. Math. 63, 619–627 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  26. Wintner, A.: On the distribution function of the remainder term of the prime number theorem. Am. J. Math. 63, 233–248 (1941)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

We thank Steve Lester and the reviewer for their comments. The work was supported by the European Research Council, under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 320755.

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Authors and Affiliations

  1. Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Bingrong Huang & Zeév Rudnick

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  1. Bingrong Huang
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  2. Zeév Rudnick
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Correspondence to Bingrong Huang.

Additional information

Communicated by A. Constantin.

Dedicated to Dorian Goldfeld on the occasion of his 71st birthday.

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Huang, B., Rudnick, Z. Prime lattice points in ovals. Monatsh Math 189, 295–319 (2019). https://doi.org/10.1007/s00605-018-1226-3

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  • DOI: https://doi.org/10.1007/s00605-018-1226-3

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