The Ramanujan Journal

, Volume 48, Issue 3, pp 495–508

Eisenstein series and convolution sums

• Zafer Selcuk Aygin
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)
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)
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)
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)
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)
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)
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)
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)
11. 11.
Cooper, S., Toh, P.C.: Quintic and septic Eisenstein series. Ramanujan J. 19, 163–181 (2009)
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)
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)
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)
17. 17.
Köhler, G.: Eta Products and Theta Series Identities. Springer Monographs in Mathematics. Springer, Heidelberg (2011)
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)
19. 19.
Ligozat, G.: Courbes modulaires de genre 1. Bull. Soc. Math. France 43, 5–80 (1975)
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)
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)
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)
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)
27. 27.
Williams, K.S.: The convolution sum $$\sum _{m{<}n/8} \sigma (m) \sigma (n-8m)$$. Pac. J. Math. 228, 387–396 (2006)
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)