# Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52

Open Mathematics (2017)

• Volume: 15, Issue: 1, page 446-458
• ISSN: 2391-5455

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## Abstract

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The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $\begin{array}{c}\sum _{\genfrac{}{}{0pt}{}{\left(l\phantom{\rule{0.166667em}{0ex}},m\right)\in {ℕ}_{0}^{2}}{\alpha \phantom{\rule{0.166667em}{0ex}}l+\beta \phantom{\rule{0.166667em}{0ex}}m=n}}\sigma \left(l\right)\sigma \left(m\right),\end{array}$ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms [...] a(x12+x22+x32+x42)+b(x52+x62+x72+x82), $a\phantom{\rule{0.166667em}{0ex}}\left({x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}+{x}_{4}^{2}\right)+b\phantom{\rule{0.166667em}{0ex}}\left({x}_{5}^{2}+{x}_{6}^{2}+{x}_{7}^{2}+{x}_{8}^{2}\right),$ where (a, b) = (1, 11), (1, 13).

## How to cite

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Ebénézer Ntienjem. "Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52." Open Mathematics 15.1 (2017): 446-458. <http://eudml.org/doc/288124>.

@article{EbénézerNtienjem2017,
abstract = {The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $\begin\{array\}\{\} \sum \limits _\{\{(l\, ,m)\in \mathbb \{N\}_\{0\}^\{2\}\}\atop \{\alpha \,l+\beta \, m=n\}\} \sigma (l)\sigma (m), \end\{array\}$ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms [...] a(x12+x22+x32+x42)+b(x52+x62+x72+x82), $a\,(x_\{1\}^\{2\}+x_\{2\}^\{2\}+x_\{3\}^\{2\}+x_\{4\}^\{2\})+b\,(x_\{5\}^\{2\}+x_\{6\}^\{2\}+x_\{7\}^\{2\}+x_\{8\}^\{2\}),$ where (a, b) = (1, 11), (1, 13).},
author = {Ebénézer Ntienjem},
journal = {Open Mathematics},
keywords = {Sums of Divisors function; Convolution Sums; Dedekind eta function; Modular Forms; Eisenstein Series; Cusp Forms; Octonary quadratic Forms; Number of Representations; sums of divisors function; convolution sums; modular forms; Eisenstein series; cusp forms; octonary quadratic forms; number of representations},
language = {eng},
number = {1},
pages = {446-458},
title = {Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52},
url = {http://eudml.org/doc/288124},
volume = {15},
year = {2017},
}

TY - JOUR
AU - Ebénézer Ntienjem
TI - Evaluation of the convolution sum involving the sum of divisors function for 22, 44 and 52
JO - Open Mathematics
PY - 2017
VL - 15
IS - 1
SP - 446
EP - 458
AB - The convolution sum, [...] ∑(l,m)∈N02αl+βm=nσ(l)σ(m), $\begin{array}{} \sum \limits _{{(l\, ,m)\in \mathbb {N}_{0}^{2}}\atop {\alpha \,l+\beta \, m=n}} \sigma (l)\sigma (m), \end{array}$ where αβ = 22, 44, 52, is evaluated for all natural numbers n. Modular forms are used to achieve these evaluations. Since the modular space of level 22 is contained in that of level 44, we almost completely use the basis elements of the modular space of level 44 to carry out the evaluation of the convolution sums for αβ = 22. We then use these convolution sums to determine formulae for the number of representations of a positive integer by the octonary quadratic forms [...] a(x12+x22+x32+x42)+b(x52+x62+x72+x82), $a\,(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2})+b\,(x_{5}^{2}+x_{6}^{2}+x_{7}^{2}+x_{8}^{2}),$ where (a, b) = (1, 11), (1, 13).
LA - eng
KW - Sums of Divisors function; Convolution Sums; Dedekind eta function; Modular Forms; Eisenstein Series; Cusp Forms; Octonary quadratic Forms; Number of Representations; sums of divisors function; convolution sums; modular forms; Eisenstein series; cusp forms; octonary quadratic forms; number of representations
UR - http://eudml.org/doc/288124
ER -

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