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Evaluation of the sums \(\displaystyle\sum_{m=1\atop m\equiv a\pmod 4}^{n-1} \sigma (m) \sigma (n-m) \). (English) Zbl 1204.11009

Summary: The convolution sum \(\displaystyle\sum_{m=1\atop m\equiv a\pmod 4}^{n-1}\sigma(m)\sigma(n-m)\) is evaluated for \(a\in\{ 0,1,2,3\}\) and all \(n\in\mathbb N\). This completes the partial evaluation given in the paper of J. G. Huard, Z. M. Ou, B. K. Spearman and K. S. Williams [Number theory for the millennium II. Proceedings of the millennial conference on number theory, Urbana-Champaign, IL, USA, 2000. Natick, MA: A. K. Peters, 229–274 (2002; Zbl 1062.11005)].

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11F27 Theta series; Weil representation; theta correspondences

Citations:

Zbl 1062.11005
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References:

[1] A. Alaca, S. Alaca, K. S. Williams: Seven octonary quadratic form. Acta Arith. 135 (2008), 339–350. · Zbl 1171.11021 · doi:10.4064/aa135-4-3
[2] B. C. Berndt: Number Theory in the Spirit of Ramanujan. American Mathematical Society (AMS), Providence, 2006. · Zbl 1117.11001
[3] N. Cheng: Convolution sums involving divisor functions. M.Sc. thesis. Carleton University, Ottawa, 2003.
[4] N. Cheng, K. S. Williams: Convolution sums involving the divisor function. Proc. Edinb. Math. Soc. 47 (2004), 561–572. · Zbl 1156.11301 · doi:10.1017/S0013091503000956
[5] J. G. Huard, Z. M. Ou, B. K. Spearman, K. S. Williams: Elementary evaluation of certain convolution sums involving divisor functions. Number Theory for the Millenium II (Urbana, IL, 2000). A.K. Peters, Natick, 2002, pp. 229–274. · Zbl 1062.11005
[6] K. S. Williams: The convolution sum \( \sum\limits_{m < n/8} {\sigma (m)\sigma (n - 8m)} \) . Pac. J. Math. 228 (2006), 387–396. · Zbl 1130.11006 · doi:10.2140/pjm.2006.228.387
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