Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-06-24T08:25:20.120Z Has data issue: false hasContentIssue false
  • English
  • Français

The distribution of consecutive prime biases and sums of sawtooth random variables

Published online by Cambridge University Press:  02 August 2018

ROBERT J. LEMKE OLIVER  and
KANNAN SOUNDARARAJAN
ROBERT J. LEMKE OLIVER
Affiliation:
Department of Mathematics, Tufts University, 503 Boston Ave. Medford, MA 02155, U.S.A. e-mail: robert.lemke_oliver@tufts.edu
KANNAN SOUNDARARAJAN
Affiliation:
Department of Mathematics, Stanford University, 450 Serra Mall, Building 380. Stanford, CA 94305-2125, U.S.A. e-mail: ksound@stanford.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In recent work, we considered the frequencies of patterns of consecutive primes (mod q) and numerically found biases toward certain patterns and against others. We made a conjecture explaining these biases, the dominant factor in which permits an easy description but fails to distinguish many patterns that have seemingly very different frequencies. There was a secondary factor in our conjecture accounting for this additional variation, but it was given only by a complicated expression whose distribution was not easily understood. Here, we study this term, which proves to be connected to both the Fourier transform of classical Dedekind sums and the error term in the asymptotic formula for the sum of φ(n).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 1 , January 2020 , pp. 149 - 169
Copyright
Copyright © Cambridge Philosophical Society 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Partially supported by NSF grant DMS-1601398.

‡

Partially supported by NSF grant DMS-1500237 and by a Simons Investigator grant from the Simons Foundation.

References

[1] Apostol, T. M. Modular functions and Dirichlet series in number theory. Graduate Texts in Mathematics. vol. 41 (Springer-Verlag, New York, second edition, 1990).Google Scholar
[2] Bourgain, J. and Garaev, M. Z. Sumsets of reciprocals in prime fields and multilinear Kloosterman sums. Izv. Ross. Akad. Nauk Ser. Mat. 78 (4) (2014), 19–72.Google Scholar
[3] Chowla, S. Contributions to the analytic theory of numbers. Math. Z. 35 (1) (1932), 279–299.Google Scholar
[4] Erdős, P. and Shapiro, H. N. The existence of a distribution function for an error term related to the Euler function. Canad. J. Math. 7 (1955), 63–75.Google Scholar
[5] Feller, W. An Introduction to Probability Theory and its Applications. Vol. II. Second edition (John Wiley & Sons, Inc., New York-London-Sydney, 1971).Google Scholar
[6] Granville, A. and Soundararajan, K. The distribution of values of L(1, χd). Geom. Funct. Anal. 13 (5) (2003), 992–1028.Google Scholar
[7] Karatsuba, A. A. New estimates for short Kloosterman sums. Mat. Zametki 88 (3) (2010), 384–398.Google Scholar
[8] Korolëv, M. A. On Karatsuba's method of estimating Kloosterman sums. Mat. Sb. 207 (8) (2016), 117–134.Google Scholar
[9] Lau, Y.-K.. On the existence of limiting distributions of some number-theoretic error terms. J. Number Theory 94 (2) (2002), 359–374.Google Scholar
[10] Lemke Oliver, R. J. and Soundararajan, K.. Unexpected biases in the distribution of consecutive primes. Proc. Natl. Acad. Sci. USA 113 (31) (2016), E4446–E4454.Google Scholar
[11] Montgomery, H. L. Fluctuations in the mean of Euler's phi function. Proc. Indian Acad. Sci. Math. Sci. 97 (1–3) (1987), 239–245.Google Scholar
[12] Montgomery, H. L. Ten lectures on the interface between analytic number theory and harmonic analysis. CBMS Regional Conference Series in Mathematics, vol. 84. Published for the Conference Board of the Mathematical Sciences, Washington, DC (American Mathematical Society, Providence, RI, 1994).Google Scholar
[13] Pillai, S. S. and Chowla, S. D. On the error terms in some asymptotic formulae in the theory of numbers (1). J. London Math. Soc. S1–5 (2) (1930), 95.Google Scholar
[14] Vardi, I. Dedekind sums have a limiting distribution. Internat. Math. Res. Not. (1), (1993), 1–12.Google Scholar
[15] Walfisz, A. Weylsche Exponentialsummen in der neueren Zahlentheorie. Mathematische Forschungsberichte, XV (VEB Deutscher Verlag der Wissenschaften, Berlin, 1963).Google Scholar
[16] Washington, L. C. Introduction to cyclotomic fields, Graduate Texts in Math. vol. 83 (Springer-Verlag, New York, second edition, 1997).Google Scholar