Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 167–198 | Cite as

Sums of divisor functions in \(\mathbb {F}_q[t]\) and matrix integrals

  • J. P. Keating
  • B. Rodgers
  • E. Roditty-Gershon
  • Z. Rudnick
Article
  • 133 Downloads

Abstract

We study the mean square of sums of the kth divisor function \(d_k(n)\) over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as \(q\rightarrow \infty \) we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of \(d_k(n)\) in terms of a lattice point count. This lattice point count can in turn be calculated in terms of a certain piecewise polynomial function, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.

References

  1. 1.
    Andrade, J.C., Bary-Soroker, L., Rudnick, Z.: Shifted convolution and the Titchmarsh divisor problem over \(\mathbb{F}_q[T]\). Philos. Trans. A 373, 20140308–20140318 (2040)CrossRefMATHGoogle Scholar
  2. 2.
    Baryshnikov, Y.: GUEs and queues. Probab. Theory Rel. Fields 119, 256–274 (2001)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Blomer, V.: The average value of divisor sums in arithmetic progressions. Q. J. Math. 59, 275–286 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bump, D.: Lie groups. Graduate texts in mathematics, vol. 225. Springer, New York (2004)Google Scholar
  5. 5.
    Bump, D., Gamburd, A.: On the averages of characteristic polynomials from classical groups Comm. Math. Phys. 265(1), 227–274 (2006)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Conrey, J.B., Gonek, S.M.: High moments of the Riemann zeta-function. Duke Math. J. 107(3), 577–604 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Autocorrelation of random matrix polynomials. Commun. Math. Phys. 237, 365–395 (2003)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Conrey, J.B., Farmer, D.W., Keating, J.P., Rubinstein, M.O., Snaith, N.C.: Integral moments of \(L\)-functions. Proc. Lond. Math. Soc. 91(3), 33–104 (2005)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Coppola, G., Salerno, S.: On the symmetry of the divisor function in almost all short intervals. Acta Arith. 113(2), 189–201 (2004)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cramér, H.: Über zwei Sätze des Herrn G. H. Hardy. Math. Z 15(1), 201–210 (1922)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Davenport, H.: On a principle of Lipschitz. J. Lond. Math. Soc. 26, 179–183 (1951)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Davenport, H.: Corrigendum: on a principle of Lipschitz. J. Lond. Math. Soc. 39, 580 (1964)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Diaconis, P., Gamburd, A.: Random matrices, magic squares and matching polynomials. Electron. J. Comb. 11(2), 26 (2004/06) (Research Paper 2)Google Scholar
  14. 14.
    Ehrhart, E.: Sur un probleme de geometrie diophantienne lineaire II. J. Reine Angew. Math. 227, 2549 (1967)Google Scholar
  15. 15.
    Fouvry, É., Ganguly, S., Kowalski, E., Michel, P.: Gaussian distribution for the divisor function and Hecke eigenvalues in arithmetic progressions. Comment. Math. Helv. 89(4), 979–1014 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fouvry, É., Ganguly, S., Kowalski, E., Michel, P.: On the exponent of distribution of the ternary divisor function. Mathematika, 1–24 (2015). doi: 10.1112/S0025579314000096, arXiv:1304.3199 [math.NT]
  17. 17.
    Gamburd, A.: Some applications of symmetric functions theory in random matrix theory. LMS Lect. Note Ser. 341, 143–170 (2007)MathSciNetMATHGoogle Scholar
  18. 18.
    Heath-Brown, D.R.: The distribution and moments of the error term in the Dirichlet divisor problem. Acta Arith. 60(4), 389–415 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ivić, A.: On the mean square of the divisor function in short intervals. J. Théor. Nombres Bordeaux 21(2), 251–261 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ivić, A.: On the divisor function and the Riemann zeta-function in short intervals. Ramanujan J. 19(2), 207–224 (2009)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jutila, M.: On the divisor problem for short intervals. Studies in honour of Arto Kustaa Salomaa on the occasion of his fiftieth birthday. Ann. Univ. Turku. Ser. A I(186), 23–30 (1984)Google Scholar
  22. 22.
    Katz, N.M.: On a question of Keating and Rudnick about primitive Dirichlet characters with squarefree conductor. Int. Math. Res. Not. 2013(14), 3221–3249 (2013)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Katz, N.M.: Witt vectors and a question of Keating and Rudnick. Int. Math. Res. Not. 2013(16), 3613–3638 (2013)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Keating, J.P.: Symmetry transitions in random matrix theory and L-Functions. Commun. Math. Phys. 281, 499–528 (2008)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Keating, J.P., Rudnick, Z.: The variance of the number of prime polynomials in short intervals and in residue classes. Int. Math. Res. Not. (2012). doi: 10.1093/imrn/rns220 MATHGoogle Scholar
  26. 26.
    Keating, J.P., Rudnick, Z.: Squarefree polynomials and Möbius values in short intervals and arithmetic progressions. Algebra Number Theory 10(2), 375–420 (2016)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Kösters, H.: On the occurrence of the sine kernel in connection with the shifted moments of the Riemann zeta function. J. Number Theory 130, 2596–2609 (2010)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Kowalski, E., Ricotta, G.: Fourier coefficients of GL(N) automorphic forms in arithmetic progressions. Geom. Funct. Anal. 24, 1229–1297 (2014)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Lau, Y.K., Zhao, L.: On a variance of Hecke eigenvalues in arithmetic progressions. J. Number Theory 132(5), 869–887 (2012)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Lester, S.: On the variance of sums of divisor functions in short intervals. Proc. Amer. Math. Soc. 144(12), 5015–5027 (2016)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Lester, S., Yesha, N.: On the distribution of the divisor function and Hecke eigenvalues. Israel J. Math. 212(1), 443–472 (2016)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Milinovich, M.B., Turnage-Butterbaugh, C.L.: Moments of products of automorphic L-functions. J. Number Theory 139, 175–204 (2014)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Motohashi, Y.: On the distribution of the divisor function in arithmetic progressions. Acta Arith. 22, 175–199 (1973)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Olshanksi, G.: Projections of orbital measures, Gelfand–Tsetlin polytopes, and splines. J. of Lie Theory 23(4), 1011–1022 (2013)MathSciNetMATHGoogle Scholar
  35. 35.
    Schmidt, W.M.: Northcott’s theorem on heights II. The quadratic case. Acta Arith. 70(4), 343–375 (1995)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Shiu, P.: A Brun–Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313, 161–170 (1980)MathSciNetMATHGoogle Scholar
  37. 37.
    Soundararajan, K.: Moments of the Riemann zeta function. Ann. Math. 170(2), 981–993 (2009)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Stanley, R.P.: Enumerative combinatorics. Volume 2, Cambridge Studies in Advanced Mathematics, vol. 62. Cambridge University Press, Cambridge (1999)Google Scholar
  39. 39.
    Titchmarsh, E.C.: The theory of the Riemann zeta-function, 2nd edn. In: Heath-Brown, D.R. (ed.) Oxford University Press, New York (1986)Google Scholar
  40. 40.
    Tong, K.C.: On divisor problems III. Acta Math. Sin. 6, 515–541 (1956)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • J. P. Keating
    • 1
  • B. Rodgers
    • 2
  • E. Roditty-Gershon
    • 1
  • Z. Rudnick
    • 3
  1. 1.School of MathematicsUniversity of BristolBristolUK
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

Personalised recommendations