Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 


The average of the divisor function over values of a quadratic polynomial

Authors: Sheng-Chi Liu and Riad Masri
Journal: Proc. Amer. Math. Soc. 143 (2015), 4143-4160
MSC (2010): Primary 11M41
Published electronically: June 5, 2015
MathSciNet review: 3373915
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We establish a uniform asymptotic formula with a power saving error term for the average of the divisor function $ \tau (n):=\sum _{k\vert n}1$ over values of the quadratic polynomial $ x^2 +\vert D\vert$ where $ D < 0$ is a fundamental discriminant.

References [Enhancements On Off] (What's this?)

  • [BM] Ehud Moshe Baruch and Zhengyu Mao, A generalized Kohnen-Zagier formula for Maass forms, J. Lond. Math. Soc. (2) 82 (2010), no. 1, 1-16. MR 2669637 (2012c:11100),
  • [Bl] Valentin Blomer, Non-vanishing of class group $ L$-functions at the central point, Ann. Inst. Fourier (Grenoble) 54 (2004), no. 4, 831-847 (English, with English and French summaries). MR 2111013 (2005m:11212)
  • [BH] Valentin Blomer and Gergely Harcos, Hybrid bounds for twisted $ L$-functions, J. Reine Angew. Math. 621 (2008), 53-79. MR 2431250 (2009e:11094),
  • [Bu] D. A. Burgess, On character sums and $ L$-series, Proc. London Math. Soc. (3) 12 (1962), 193-206. MR 0132733 (24 #A2570)
  • [By] V. A. Bykovskiĭ, Spectral expansions of certain automorphic functions and their number-theoretic applications, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 134 (1984), 15-33 (Russian, with English summary). Automorphic functions and number theory, II. MR 741852 (86g:11031)
  • [CI] J. B. Conrey and H. Iwaniec, The cubic moment of central values of automorphic $ L$-functions, Ann. of Math. (2) 151 (2000), no. 3, 1175-1216. MR 1779567 (2001g:11070),
  • [DI] J.-M. Deshouillers and H. Iwaniec, Kloosterman sums and Fourier coefficients of cusp forms, Invent. Math. 70 (1982/83), no. 2, 219-288. MR 684172 (84m:10015),
  • [DFI] W. Duke, J. Friedlander, and H. Iwaniec, Class group $ L$-functions, Duke Math. J. 79 (1995), no. 1, 1-56. MR 1340293 (97h:11138),
  • [DFI2] W. Duke, J. B. Friedlander, and H. Iwaniec, Weyl sums for quadratic roots, Int. Math. Res. Not. IMRN 11 (2012), 2493-2549. MR 2926988,
  • [FI] John Friedlander and Henryk Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, vol. 57, American Mathematical Society, Providence, RI, 2010. MR 2647984 (2011d:11227)
  • [FI2] J. B. Friedlander and H. Iwaniec, Small representations by indefinite ternary quadratic forms, Number theory and related fields, Springer Proc. Math. Stat., vol. 43, Springer, New York, 2013, pp. 157-164. MR 3081039,
  • [HL] Jeffrey Hoffstein and Paul Lockhart, Coefficients of Maass forms and the Siegel zero, Ann. of Math. (2) 140 (1994), no. 1, 161-181. With an appendix by Dorian Goldfeld, Hoffstein and Daniel Lieman. MR 1289494 (95m:11048),
  • [H] Christopher Hooley, On the number of divisors of a quadratic polynomial, Acta Math. 110 (1963), 97-114. MR 0153648 (27 #3610)
  • [I] Henryk Iwaniec, Small eigenvalues of Laplacian for $ \Gamma _0(N)$, Acta Arith. 56 (1990), no. 1, 65-82. MR 1067982 (92h:11045)
  • [IK] Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53, American Mathematical Society, Providence, RI, 2004. MR 2061214 (2005h:11005)
  • [KS] Svetlana Katok and Peter Sarnak, Heegner points, cycles and Maass forms, Israel J. Math. 84 (1993), no. 1-2, 193-227. MR 1244668 (94h:11051),
  • [K] Henry H. Kim, Functoriality for the exterior square of $ {\rm GL}_4$ and the symmetric fourth of $ {\rm GL}_2$, J. Amer. Math. Soc. 16 (2003), no. 1, 139-183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR 1937203 (2003k:11083),
  • [KZ] W. Kohnen and D. Zagier, Values of $ L$-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, 175-198. MR 629468 (83b:10029),
  • [L] Serge Lang, Elliptic functions, 2nd ed., Graduate Texts in Mathematics, vol. 112, Springer-Verlag, New York, 1987. With an appendix by J. Tate. MR 890960 (88c:11028)
  • [M] Riad Masri, The asymptotic distribution of traces of cycle integrals of the $ j$-function, Duke Math. J. 161 (2012), no. 10, 1971-2000. MR 2954622,
  • [Te] Nicolas Templier, Heegner points and Eisenstein series, Forum Math. 23 (2011), no. 6, 1135-1158. MR 2855045,
  • [Te2] Nicolas Templier, A nonsplit sum of coefficients of modular forms, Duke Math. J. 157 (2011), no. 1, 109-165. MR 2783929 (2012h:11111),
  • [TT] Nicolas Templier and Jacob Tsimerman, Non-split sums of coefficients of $ GL(2)$-automorphic forms, Israel J. Math. 195 (2013), no. 2, 677-723. MR 3096570,
  • [T] E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550 (88c:11049)
  • [Z] D. Zagier, Modular forms whose Fourier coefficients involve zeta-functions of quadratic fields, Modular functions of one variable, VI (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976) Springer, Berlin, 1977, pp. 105-169. Lecture Notes in Math., Vol. 627. MR 0485703 (58 #5525)
  • [Z2] D. Zagier, Eisenstein series and the Riemann zeta function, Automorphic forms, representation theory and arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., vol. 10, Tata Inst. Fundamental Res., Bombay, 1981, pp. 275-301. MR 633666 (83j:10027)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11M41

Retrieve articles in all journals with MSC (2010): 11M41

Additional Information

Sheng-Chi Liu
Affiliation: Department of Mathematics, Washington State University, Pullman, Washington 99164-3113

Riad Masri
Affiliation: Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, Texas 77843-3368

Received by editor(s): October 16, 2013
Received by editor(s) in revised form: March 3, 2014
Published electronically: June 5, 2015
Communicated by: Ken Ono
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society