Evaluation of Convolution Sums entailing mixed Divisor Functions for a Class of Levels

Authors

  • Ebenezer Ntienjem Carleton University

Keywords:

Sums of Divisors, Dedekind eta function, Convolution Sums, Modular Forms, Dirichlet Characters, Eisenstein forms, Cusp Forms, Octonary quadratic Forms, Number of Representations

Abstract

Let $0< n,\alpha,\beta\in\mathbb{N}$ be such that $\gcd{(\alpha,\beta)}=1$. We carry out the evaluation of the convolution sums $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {\alpha\,k+\beta\,l=n} } }{\sum}\sigma(k)\sigma_{3}(l)$ and $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {\alpha\,k+\beta\,l=n} } }{\sum}\sigma_{3}(k)\sigma(l)$ for all levels $\alpha\beta\in\mathbb{N}$, by using in particular modular forms. We next apply convolution sums belonging to this class of levels to determine formulae for the number of representations of a positive integer $n$ by the quadratic forms in twelve variables $\underset{i=1}{\overset{12}{\sum}}x_{i}^{2}$ when the level $\alpha\beta\equiv 0\pmod{4}$, and $\underset{i=1}{\overset{6}{\sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,)$ when the level $\alpha\beta\equiv 0\pmod{3}$. Our approach is then illustrated by explicitly evaluating the convolution sum for $\alpha\beta=3$, $6$, $7$, $8$, $9$, $12$, $14$, $15$, $16$, $18$, $20$, $21$, $27$, $32$. These convolution sums are then applied to determine explicit formulae for the number of representations of a positive integer $n$ by quadratic forms in twelve variables.

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Author Biography

Ebenezer Ntienjem, Carleton University

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

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Published

2021-02-04

How to Cite

Ntienjem, E. (2021). Evaluation of Convolution Sums entailing mixed Divisor Functions for a Class of Levels. New Zealand Journal of Mathematics, 50, 125–180. Retrieved from https://nzjmath.org/index.php/NZJMATH/article/view/80

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