| Title:
|
Evaluation of the sums $\sum\limits_{\substack{m=1 \\ m\equiv a\pmod 4}}^{n-1} \sigma (m) \sigma (n-m) $ (English) |
| Author:
|
Alaca, Ayşe |
| Author:
|
Alaca, Şaban |
| Author:
|
Williams, Kenneth S. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
59 |
| Issue:
|
3 |
| Year:
|
2009 |
| Pages:
|
847-859 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
convolution sums |
| Keyword:
|
sum of divisors function |
| Keyword:
|
theta functions |
| MSC:
|
11A25 |
| MSC:
|
11F27 |
| idZBL:
|
Zbl 1204.11009 |
| idMR:
|
MR2545660 |
| . |
| Date available:
|
2010-07-20T15:43:40Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140520 |
| . |
| Reference:
|
[1] Alaca, A., Alaca, S., Williams, K. S.: Seven octonary quadratic form.Acta Arith. 135 (2008), 339-350. MR 2465716, 10.4064/aa135-4-3 |
| Reference:
|
[2] Berndt, B. C.: Number Theory in the Spirit of Ramanujan.American Mathematical Society (AMS) Providence (2006). Zbl 1117.11001, MR 2246314 |
| Reference:
|
[3] Cheng, N.: Convolution sums involving divisor functions.M.Sc. thesis Carleton University Ottawa (2003). |
| Reference:
|
[4] Cheng, N., Williams, K. S.: Convolution sums involving the divisor function.Proc. Edinb. Math. Soc. 47 (2004), 561-572. Zbl 1156.11301, MR 2096620, 10.1017/S0013091503000956 |
| Reference:
|
[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. Zbl 1062.11005, MR 1956253 |
| Reference:
|
[6] Williams, K. S.: The convolution sum $\sum_{m< n/8} \sigma(m) \sigma(n-8m)$.Pac. J. Math. 228 (2006), 387-396. Zbl 1130.11006, MR 2274527 |
| . |
