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

• Autorzy: Ebénézer Ntienjem
• Języki publikacji: EN

Abstrakty

EN

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).

Słowa kluczowe

EN

Sums of Divisors function; Convolution Sums; Dedekind eta function; Modular Forms; Eisenstein Series; Cusp Forms; Octonary quadratic Forms; Number of Representations

Kategorie tematyczne

• 11A25: Arithmetic functions; related numbers; inversion formulas
• 11E20: General ternary and quaternary quadratic forms; forms of more than two variables
• 11F27: Theta series; Weil representation; theta correspondences
• 11F20: Dedekind eta function, Dedekind sums
• 11F11: Holomorphic modular forms of integral weight
• 11E25: Sums of squares and representations by other particular quadratic forms

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2017
• Tom: 15
• Numer: 1
• Strony: 446-458

Daty

wydano

2017-01-01

otrzymano

2016-08-30

zaakceptowano

2017-02-28

online

2017-04-18

Twórcy

autor

Ebénézer Ntienjem

• Centre for Research in Algebra and Number Theory, School of Mathematics and Statistics, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, K1S 5B6,

Identyfikatory

DOI

10.1515/math-2017-0041