The Ramanujan Journal

, Volume 48, Issue 3, pp 495–508 | Cite as

Eisenstein series and convolution sums

  • Zafer Selcuk AyginEmail author


We compute Fourier series expansions of weight 2 and weight 4 Eisenstein series at various cusps. Then we use results of these computations to give formulas for the convolution sums \( \sum _{a+p b=n}\sigma (a)\sigma (b)\), \( \sum _{p_1a+p_2 b=n}\sigma (a)\sigma (b)\) and \( \sum _{a+p_1 p_2 b=n}\sigma (a)\sigma (b)\) where \(p, p_1, p_2\) are primes.


Sum of divisors function Convolution sums Eisenstein series Dedekind eta function Eta quotients Modular forms Cusp forms Fourier series 

Mathematics Subject Classification

11A25 11E20 11F11 11F20 11F30 11Y35 



I would like to thank the anonymous referee for his very useful comments.


  1. 1.
    Alaca, A., Alaca, Ş., Williams, K.S.: Evaluation of the convolution sums \(\sum _{l+12m=n} \sigma (l) \sigma (m) \) and \(\sum _{3l+4m=n} \sigma (l) \sigma (m) \). Adv. Theor. Appl. Math. 1, 27–48 (2006)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Alaca, A., Alaca, Ş., Williams, K.S.: Evaluation of the convolution sums \(\sum _{l+18m=n} \sigma (l) \sigma (m) \) and \(\sum _{2l+9m=n} \sigma (l) \sigma (m) \). Int. Math. Forum 2, 45–68 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Alaca, A., Alaca, Ş., Williams, K.S.: Evaluation of the convolution sums \(\sum _{l+24m=n} \sigma (l) \sigma (m) \) and \(\sum _{3l+8m=n} \sigma (l) \sigma (m) \). Math. J. Okayama Univ. 49, 93–111 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Alaca, A., Alaca, Ş., Williams, K.S.: The convolution sum \(\sum _{m{<}n/16} \sigma (m)\sigma (n - 16m)\). Can. Math. Bull. 51, 3–14 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Alaca, Ş., Kesicioğlu, Y.: Evaluation of the convolution sums \(\sum _{l+27m=n} \sigma (l) \sigma (m) \) and \(\sum _{l+32m=n} \sigma (l) \sigma (m) \). J. Number Theory 1, 1–13 (2016)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Alaca, Ş., Williams, K.S.: Evaluation of the convolution sums \(\sum _{l+6m=n} \sigma (l) \sigma (m) \) and \(\sum _{2l+3m=n} \sigma (l) \sigma (m) \). J. Number Theory 124, 491–510 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Aygin, Z.S.: Eisenstein series, eta quotients and their applications in number theory. Doctoral dissertation, Carleton University, Ottawa, Canada (2016)Google Scholar
  8. 8.
    Aygin, Z.S.: Extensions of Ramanujan-Mordell formula with coefficients \(1\) and \(p\). J. Math. Anal. Appl. 465, 690–702 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Besge, M.: Extrait d’une lettre de M. Besge \(\acute{a}\) M. Liouville. J. Math. Pures Appl. 7, 256 (1862)Google Scholar
  10. 10.
    Chan, H.H., Cooper, S.: Powers of theta functions. Pac. J. Math. 235, 1–14 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cooper, S., Toh, P.C.: Quintic and septic Eisenstein series. Ramanujan J. 19, 163–181 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cooper, S., Ye, D.: Evaluation of the convolution sums \(\sum _{l+20m=n} \sigma (l) \sigma (m) \), \(\sum _{4l+5m=n} \sigma (l) \sigma (m) \) and \(\sum _{2l+5m=n} \sigma (l) \sigma (m) \). Int. J. Number Theory 6, 1385–1394 (2014)CrossRefzbMATHGoogle Scholar
  13. 13.
    Glaisher, J.W.L.: On the square of the series in which the coefficients are the sums of the divisors of the exponents. Mess. Math. 14, 156–163 (1885)Google Scholar
  14. 14.
    Gordon, B., Sinor, D.: Multiplicative properties of \(\eta \)-products. In: Alladi, K. (ed.) Number Theory, Madras 1987. Lecture Notes in Mathematics, vol. 1395, pp. 173–200. Springer, New York (1989)CrossRefGoogle Scholar
  15. 15.
    Huard, J.G., Ou, Z.M., Spearman, B.K., Williams, K.S.: Elementary evaluation of certain convolution sums involving divisor functions. In: Peters, A.K. (ed.) Number Theory for the Millennium, II, pp. 229–274. Natick, MA (2002)Google Scholar
  16. 16.
    Kilford, L.J.P.: Modular Forms. A Classical and Computational Introduction. Imperial College Press, London (2008)CrossRefzbMATHGoogle Scholar
  17. 17.
    Köhler, G.: Eta Products and Theta Series Identities. Springer Monographs in Mathematics. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  18. 18.
    Lemire, M., Williams, K.S.: Evaluation of two convolution sums involving the sum of divisor functions. Bull. Aust. Math. Soc. 73, 107–115 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Ligozat, G.: Courbes modulaires de genre 1. Bull. Soc. Math. France 43, 5–80 (1975)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Ntienjem, E.: Evaluation of the convolution sum involving the sum of divisors function for \(22\), \(44\) and \(52\). Open Math. 15, 446–458 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Ramakrishnan, B., Sahu, B.: Evaluation of the convolution sums \(\sum _{l+15m=n} \sigma (l) \sigma (m) \) and \(\sum _{3l+5m=n} \sigma (l) \sigma (m) \) and an application. Int. J. Number Theory 9, 799–809 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Ramanujan, S.: On certain arithmetical functions. Trans. Camb. Philos. Soc. 22, 159–184 (1916)Google Scholar
  23. 23.
    Royer, E.: Evaluating convolution sums of the divisor function by quasimodular forms. Int. J. Number Theory 3, 231–261 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Serre, J.-P.: Modular forms of weight one and Galois representations, Algebraic number fields: L-functions and Galois properties. In: Proceedings of the Durham Symposium, University of Durham, Durham, 1975, pp. 193–268 . Academic Press. London (1977)Google Scholar
  25. 25.
    Stein, W.A.: Modular Forms, A Computational Approach. Graduate Studies in Mathematics, vol. 79. American Mathematical Society, Providence (2007)Google Scholar
  26. 26.
    Williams, K.S.: The convolution sum \(\sum _{m{<}n/9} \sigma (m) \sigma (n-9m) \). Int. J. Number Theory 1, 193–205 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Williams, K.S.: The convolution sum \(\sum _{m{<}n/8} \sigma (m) \sigma (n-8m) \). Pac. J. Math. 228, 387–396 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Xia, E.X.W., Tian, X.L., Yao, O.X.M.: Evaluation of the convolution sum \(\sum _{l+25m=n} \sigma (l) \sigma (m) \). Int. J. Number Theory 10, 1421 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore

Personalised recommendations