Williams, Kenneth S. The convolution sum \(\sum_{m<n/9}\sigma(m)\sigma(n-9m)\). (English) Zbl 1082.11003 Int. J. Number Theory 1, No. 2, 193-205 (2005). Let \(\sigma_j(n)= \sum_{d|n} d^j\) and \(W_k(n)= \sum_{0< m<{n\over k}}\sigma(m) \sigma(n- km)\). Liouville has proved \(W_1(n)= {5\over 12}\sigma_3(n)+ {1- 6n\over 12}\sigma_1(n)\) for all positive integers \(n\). Similar but more complicated formulae are known for \(k= 2,3,4,5\) and \(9\). The author proves a formula for \(W_9(n)\), which contains the known one. He uses Eisenstein-series, Ramanujan’s tau-function and their relations. Reviewer: Jürgen Spilker (Freiburg i. Br.) Cited in 29 Documents MSC: 11A25 Arithmetic functions; related numbers; inversion formulas 11E20 General ternary and quaternary quadratic forms; forms of more than two variables 11E25 Sums of squares and representations by other particular quadratic forms Keywords:divisor sums; arithmetical functions; Eisenstein series PDFBibTeX XMLCite \textit{K. S. Williams}, Int. J. Number Theory 1, No. 2, 193--205 (2005; Zbl 1082.11003) Full Text: DOI References: [1] DOI: 10.1007/978-1-4612-4530-8 · doi:10.1007/978-1-4612-4530-8 [2] DOI: 10.1007/978-1-4612-1624-7 · doi:10.1007/978-1-4612-1624-7 [3] Berndt B. C., Trans. Amer. Math. Soc. 347 pp 4163– [4] Besge M., J. Math. Pures Appl. 7 pp 256– [5] Copson E. T., An Introduction to the Theory of Functions of a Complex Variable (1960) · Zbl 0188.37901 [6] Glaisher J. W. L., Mess. Math. 14 pp 156– [7] Glaisher J. W. L., Mathematical Papers, 1883–1885 (1885) [8] J. G. Huard, Number Theory for the Millennium II, eds. M. A. Bennett (A. K. Peters, Natick, Massachusetts, 2002) pp. 229–274. [9] Lahiri D. B., Bull. Calcutta Math. Soc. 38 pp 193– [10] Lahiri D. B., Bull. Calcutta Math. Soc. 39 pp 33– [11] Lützen J., Joseph Liouville 1809–1882: Master of Pure and Applied Mathematics (1998) [12] DOI: 10.1515/9783110809794.371 · doi:10.1515/9783110809794.371 [13] Mordell L. J., Proc. Cambridge Philos. Soc. 19 pp 117– [14] Ramanujan S., Trans. Cambridge Phil. Soc. 22 pp 159– [15] Ramanujan S., Collected Papers (2000) [16] DOI: 10.1017/S0305004104007832 · doi:10.1017/S0305004104007832 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.