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convolution sums; sum of divisors function; theta functions
The convolution sum $$ \sum\limits_{\substack{m=1 \\ m\equiv a\pmod 4}}^{n-1} \sigma (m) \sigma (n-m) $$ is evaluated for $a\in \{ 0,1,2,3\}$ and all $n \in \Bbb N$. This completes the partial evaluation given in the paper of J. G. Huard, Z. M. Ou, B. K. Spearman, K. S. Williams.
[1] Alaca, A., Alaca, S., Williams, K. S.: Seven octonary quadratic form. Acta Arith. 135 (2008), 339-350. DOI 10.4064/aa135-4-3 | MR 2465716
[2] Berndt, B. C.: Number Theory in the Spirit of Ramanujan. American Mathematical Society (AMS) Providence (2006). MR 2246314 | Zbl 1117.11001
[3] Cheng, N.: Convolution sums involving divisor functions. M.Sc. thesis Carleton University Ottawa (2003).
[4] Cheng, N., Williams, K. S.: Convolution sums involving the divisor function. Proc. Edinb. Math. Soc. 47 (2004), 561-572. DOI 10.1017/S0013091503000956 | MR 2096620 | Zbl 1156.11301
[5] Huard, J. G., Ou, Z. M., Spearman, B. K., Williams, K. S.: Elementary evaluation of certain convolution sums involving divisor functions. Number Theory for the Millenium II (Urbana, IL, 2000) A. K. Peters Natick (2002), 229-274. MR 1956253 | Zbl 1062.11005
[6] Williams, K. S.: The convolution sum $\sum_{m< n/8} \sigma(m) \sigma(n-8m)$. Pac. J. Math. 228 (2006), 387-396. MR 2274527 | Zbl 1130.11006
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