Evaluation of the sums $\sum\limits_{\substack{m=1 \\ m\equiv a\pmod 4}}^{n-1} \sigma (m) \sigma (n-m) $

Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S.

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Evaluation of the sums $\sum\limits_{\substack{m=1 \\ m\equiv a\pmod 4}}^{n-1} \sigma (m) \sigma (n-m) $

Alaca, Ayşe ; Alaca, Şaban ; Williams, Kenneth S.

Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 847-859.

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

Résumé

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.

MR Zbl

Classification : 11A25, 11F27
Keywords: convolution sums; sum of divisors function; theta functions

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     title = {Evaluation of the sums $\sum\limits_{\substack{m=1 \\ m\equiv a\pmod 4}}^{n-1} \sigma (m) \sigma (n-m) $},
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Alaca, Ayşe; Alaca, Şaban; Williams, Kenneth S. Evaluation of the sums $\sum\limits_{\substack{m=1 \\ m\equiv a\pmod 4}}^{n-1} \sigma (m) \sigma (n-m) $. Czechoslovak Mathematical Journal, Tome 59 (2009) no. 3, pp. 847-859. http://geodesic.mathdoc.fr/item/CMJ_2009__59_3_a20/