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


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.


  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)CrossRefzbMATHGoogle Scholar
  2. 2.
    Baryshnikov, Y.: GUEs and queues. Probab. Theory Rel. Fields 119, 256–274 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Blomer, V.: The average value of divisor sums in arithmetic progressions. Q. J. Math. 59, 275–286 (2008)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Conrey, J.B., Gonek, S.M.: High moments of the Riemann zeta-function. Duke Math. J. 107(3), 577–604 (2001)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cramér, H.: Über zwei Sätze des Herrn G. H. Hardy. Math. Z 15(1), 201–210 (1922)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Davenport, H.: On a principle of Lipschitz. J. Lond. Math. Soc. 26, 179–183 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Davenport, H.: Corrigendum: on a principle of Lipschitz. J. Lond. Math. Soc. 39, 580 (1964)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Ivić, A.: On the divisor function and the Riemann zeta-function in short intervals. Ramanujan J. 19(2), 207–224 (2009)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Katz, N.M.: Witt vectors and a question of Keating and Rudnick. Int. Math. Res. Not. 2013(16), 3613–3638 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Keating, J.P.: Symmetry transitions in random matrix theory and L-Functions. Commun. Math. Phys. 281, 499–528 (2008)MathSciNetCrossRefzbMATHGoogle 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 zbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kowalski, E., Ricotta, G.: Fourier coefficients of GL(N) automorphic forms in arithmetic progressions. Geom. Funct. Anal. 24, 1229–1297 (2014)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Milinovich, M.B., Turnage-Butterbaugh, C.L.: Moments of products of automorphic L-functions. J. Number Theory 139, 175–204 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Motohashi, Y.: On the distribution of the divisor function in arithmetic progressions. Acta Arith. 22, 175–199 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Olshanksi, G.: Projections of orbital measures, Gelfand–Tsetlin polytopes, and splines. J. of Lie Theory 23(4), 1011–1022 (2013)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Schmidt, W.M.: Northcott’s theorem on heights II. The quadratic case. Acta Arith. 70(4), 343–375 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Shiu, P.: A Brun–Titchmarsh theorem for multiplicative functions. J. Reine Angew. Math. 313, 161–170 (1980)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Soundararajan, K.: Moments of the Riemann zeta function. Ann. Math. 170(2), 981–993 (2009)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetzbMATHGoogle 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

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