Some arithmetic identities involving divisor functions. (English) Zbl 1318.11001
Summary: For a positive integer \(n\), let \(\sigma(n):= \displaystyle\sum_{d \in \mathbb{N},\ d\mid n} d\). The explicit evaluation of such arithmetic sums as
\[ \sum_{\substack{(a,b,c) \in \mathbb N^3,\\ a+2b+4c=n}} \sigma(a)\sigma(b) \sigma(c)\quad\text{ and } \sum_{\substack{ (a,b) \in \mathbb N^2,\\ a+2b=n}} a \sigma(a)\sigma(b) \]
is carried out for all positive integers \(n\).
\[ \sum_{\substack{(a,b,c) \in \mathbb N^3,\\ a+2b+4c=n}} \sigma(a)\sigma(b) \sigma(c)\quad\text{ and } \sum_{\substack{ (a,b) \in \mathbb N^2,\\ a+2b=n}} a \sigma(a)\sigma(b) \]
is carried out for all positive integers \(n\).
MSC:
| 11A25 | Arithmetic functions; related numbers; inversion formulas |
| 11F27 | Theta series; Weil representation; theta correspondences |
References:
| [1] | B. C. Berndt, Number Theory in the Spirit of Ramanujan , Amer. Math. Soc., Providence, Rhode Island, 2006. · Zbl 1117.11001 |
| [2] | N. Cheng and K. S. Williams, Evaluation of some convolution sums involving the sum of divisor functions, Yokohama Math. J. 52 (2005), 39-57. · Zbl 1094.11004 |
| [3] | {J. G. Huard, Z. M. Ou, B. K. Spearman and K. S. Williams, Elementary evaluation of certain convolution sums involving divisor functions , Number Theory for the Millenium II, edited by M. A. Bennett, B. C. Berndt, N. Boston, H. G. Diamond, A. J. Hildebrand and W. Philipp , A K Peters, Natick, Massachusetts, USA, 2002, pp. 229-274.} · Zbl 1062.11005 |
| [4] | D. B. Lahiri, On Ramanujan’s function \(\tau(n)\) and the divisor function \(\sigma_k(n)\)-I , Bull. Calcutta Math. Soc. 38 (1946), 193-206. · Zbl 0060.10203 |
| [5] | S. Ramanujan, On certain arithmetical functions , Trans. Cambridge Philos. Soc. 22 (1916), 159-184. |
| [6] | S. Ramanujan, Collected Papers , AMS Chelsea Publishing, Providence, Rhode Island, USA, 2000. · Zbl 1110.11001 |
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