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The Ramanujan Journal

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

Eisenstein series and convolution sums

  • Zafer Selcuk AyginEmail author
Article

Abstract

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.

Keywords

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 

Notes

Acknowledgements

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

References

  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

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