×

zbMATH — the first resource for mathematics

Evaluating convolution sums of the divisor function by quasimodular forms. (English) Zbl 1142.11027
Let \(W_N(n)=\sum_{m<n/N}\sigma_1(m)\sigma_1(n-Nm)\), where \(\sigma_1\) is the usual sum of divisors function. Many authors including S. Ramanujan have evaluated certain values of \(W_N(n)\) and related convolution sums. The present author provides a systematic method based on quasimodular forms to evaluate these kind of sums. This extension of modular forms has been constructed by M. Kaneko and D. Zagier [“A generalized Jacobi theta function and quasimodular forms”, Prog. Math. 129, 165–172 (1995; Zbl 0892.11015)].

MSC:
11A25 Arithmetic functions; related numbers; inversion formulas
11F11 Holomorphic modular forms of integral weight
11F20 Dedekind eta function, Dedekind sums
11F25 Hecke-Petersson operators, differential operators (one variable)
Software:
Magma
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Alaca A., Adv. Theor. Appl. Math. 1 pp 27–
[2] Alaca A., Math. J. Okayama Univ. 49
[3] Alaca A., Int. Math. Forum 2 pp 45– · Zbl 1163.11003
[4] Besge M., J. Math. Pures Appl. 7 pp 256–
[5] DOI: 10.1006/jsco.1996.0125 · Zbl 0898.68039
[6] Cheng N., Yokohama Math. J. 52 pp 39–
[7] F. Diamond and J. Im, Seminar on Fermat’s Last Theorem (Toronto, 1993–1994), CMS Conf. Proc. 17 (Amer. Math. Soc., Providence, RI, 1995) pp. 39–133.
[8] Diamond F., Graduate Texts in Mathematics 228, in: A First Course in Modular Forms (2005)
[9] Glaisher J. W. L., Mess. Math. pp 156–
[10] J. G. Huard, Number Theory for the Millennium, II (Urbana, IL, 2000) (A. K. Peters, Natick, MA, 2002) pp. 229–274.
[11] DOI: 10.1007/978-1-4612-4264-2_6
[12] Koike M., Nagoya Math. J. 95 pp 85– · Zbl 0548.10018
[13] Lahiri D. B., Bull. Calcutta Math. Soc. 38 pp 193–
[14] Lahiri D. B., Bull. Calcutta Math. Soc. 39 pp 33–
[15] Lelièvre S., Int. Math. Res. Not. 2006 pp 30–
[16] DOI: 10.1017/S0004972700038661 · Zbl 1163.11301
[17] F. Martin and E. Royer, Formes Modulaires et Transcendance, Sémin. Congr. 12 (Soc. Math. France, Paris, 2005) pp. 1–117.
[18] DOI: 10.1515/9783110809794.371
[19] Ramanujan K. S., Trans. Cambridge Philos. Soc. 22 pp 159–
[20] Rankin R. A., J. Indian Math. Soc. (N.S.) 20 pp 103–
[21] DOI: 10.2307/1970859 · Zbl 0255.10032
[22] Shimura G., Publications of the Mathematical Society of Japan 11, in: Introduction to the Arithmetic Theory of Automorphic Functions (1994)
[23] DOI: 10.1090/gsm/079
[24] DOI: 10.1142/S1793042105000091 · Zbl 1082.11003
[25] DOI: 10.2140/pjm.2006.228.387 · Zbl 1130.11006
[26] DOI: 10.1007/978-3-662-02838-4_4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.