Evaluation of the sums ∑ m = 1 m ≡ a ( mod 4 ) n - 1 σ ( m ) σ ( n - m )

Ayşe Alaca; Şaban Alaca; Kenneth S. Williams

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 3, page 847-859
  • ISSN: 0011-4642

Abstract

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The convolution sum ∑ m = 1 m ≡ a ( mod 4 ) n - 1 σ ( m ) σ ( n - m ) is evaluated for a ∈ { 0 , 1 , 2 , 3 } and all n ∈ ℕ . This completes the partial evaluation given in the paper of J. G. Huard, Z. M. Ou, B. K. Spearman, K. S. Williams.

How to cite

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Alaca, Ayşe, Alaca, Şaban, and Williams, Kenneth S.. "Evaluation of the sums $\sum \limits _{\begin{array}{c}m=1 \\ m\equiv a\hspace{4.44443pt}(\@mod \; 4)\end{array}}^{n-1} \sigma (m) \sigma (n-m) $." Czechoslovak Mathematical Journal 59.3 (2009): 847-859. <http://eudml.org/doc/37962>.

@article{Alaca2009,
abstract = {The convolution sum \[ \sum \limits \_\{\begin\{array\}\{c\}m=1 \\ m\equiv a\hspace\{10.0pt\}(\@mod \; 4)\end\{array\}\}^\{n-1\} \sigma (m) \sigma (n-m) \] is evaluated for $a\in \lbrace 0,1,2,3\rbrace $ 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, K. S. Williams.},
author = {Alaca, Ayşe, Alaca, Şaban, Williams, Kenneth S.},
journal = {Czechoslovak Mathematical Journal},
keywords = {convolution sums; sum of divisors function; theta functions; convolution sum; sum of divisors function; theta function},
language = {eng},
number = {3},
pages = {847-859},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Evaluation of the sums $\sum \limits _\{\begin\{array\}\{c\}m=1 \\ m\equiv a\hspace\{4.44443pt\}(\@mod \; 4)\end\{array\}\}^\{n-1\} \sigma (m) \sigma (n-m) $},
url = {http://eudml.org/doc/37962},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Alaca, Ayşe
AU - Alaca, Şaban
AU - Williams, Kenneth S.
TI - Evaluation of the sums $\sum \limits _{\begin{array}{c}m=1 \\ m\equiv a\hspace{4.44443pt}(\@mod \; 4)\end{array}}^{n-1} \sigma (m) \sigma (n-m) $
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 3
SP - 847
EP - 859
AB - The convolution sum \[ \sum \limits _{\begin{array}{c}m=1 \\ m\equiv a\hspace{10.0pt}(\@mod \; 4)\end{array}}^{n-1} \sigma (m) \sigma (n-m) \] is evaluated for $a\in \lbrace 0,1,2,3\rbrace $ 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, K. S. Williams.
LA - eng
KW - convolution sums; sum of divisors function; theta functions; convolution sum; sum of divisors function; theta function
UR - http://eudml.org/doc/37962
ER -

References

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  1. Alaca, A., Alaca, S., Williams, K. S., 10.4064/aa135-4-3, Acta Arith. 135 (2008), 339-350. (2008) MR2465716DOI10.4064/aa135-4-3
  2. Berndt, B. C., Number Theory in the Spirit of Ramanujan, American Mathematical Society (AMS) Providence (2006). (2006) Zbl1117.11001MR2246314
  3. Cheng, N., Convolution sums involving divisor functions, M.Sc. thesis Carleton University Ottawa (2003). (2003) 
  4. Cheng, N., Williams, K. S., 10.1017/S0013091503000956, Proc. Edinb. Math. Soc. 47 (2004), 561-572. (2004) Zbl1156.11301MR2096620DOI10.1017/S0013091503000956
  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. (2002) Zbl1062.11005MR1956253
  6. Williams, K. S., The convolution sum ∑ m < n / 8 σ ( m ) σ ( n - 8 m ) , Pac. J. Math. 228 (2006), 387-396. (2006) Zbl1130.11006MR2274527

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