arXiv subjects

Ebénézer Ntienjem

Publications and source records attributed to Ebénézer Ntienjem.

6 recordsLinked to original sources

arXiv2019

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

Let $0< n,α,β\in\mathbb{N}$ be such that $\gcd{(α,β)}=1$. We carry out the evaluation of the convolution sums $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {α\,k+β\,l=n} } }{\sum}σ(k)σ_{3}(l)$ and $\underset{\substack{ {(k,l)\in\mathbb{N}^{2}} \\ {α\,k+β\,l=n} } }{\sum}σ_{3}(k)σ(l)$ for all levels $αβ\in\mathbb{N}$, 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 $αβ\equiv 0\pmod{4}$, and $\underset{i=1}{\overset{6}{\sum}}\,(\,x_{2i-1}^{2}+ x_{2i-1}x_{2i} + x_{2i}^{2}\,)$ when the level $αβ\equiv 0\pmod{3}$. Our approach is then illustrated by explicitly evaluating the convolution sum for $αβ=3$, $4$, $6$, $7$, $8$, $9$, $12$, $14$, $15$, $16$, $18$, $20$, $21$, $27$, $32$. 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.

math.NT↗

arXiv2017

Elementary Evaluation of Convolution Sums involving the Sum of Divisors Function for a Class of positive Integers

We discuss an elementary method for the evaluation of the convolution sums $\underset{\substack{ {(l,m)\in\mathbb{N}_{0}^{2}} \\ {α\,l+β\,m=n} } }{\sum}σ(l)σ(m)$ for those $α,β\in\mathbb{N}$ for which $\gcd{(α,β)}=1$ and $αβ=2^ν\mho$, where $ν\in\{0,1,2,3\}$ and $\mho$ is a finite product of distinct odd primes. Modular forms are used to achieve this result. We also generalize the extraction of the convolution sum to all natural numbers. Formulae for the number of representations of a positive integer $n$ by octonary quadratic forms using convolution sums belonging to this class are then determined when $αβ\equiv 0\pmod{4}$ or $αβ\equiv 0\pmod{3}$. To achieve this application, we first discuss a method to compute all pairs $(a,b),(c,d)\in\mathbb{N}^{2}$ necessary for the determination of such formulae for the number of representations of a positive integer $n$ by octonary quadratic forms when $αβ$ has the above form and $αβ\equiv 0\pmod{4}$ or $αβ\equiv 0\pmod{3}$. We illustrate our approach by explicitly evaluating the convolution sum for $αβ=33=3\cdot 11,\> αβ=40=2^{3}\cdot 5$ and $αβ=56=2^{3}\cdot 7$, and by revisiting the evaluation of the convolution sums for $αβ=10$, $11$, $12$, $15$, $24$. We then apply these convolution sums to determine formulae for the number of representations of a positive integer $n$ by octonary quadratic forms. In addition, we determine formulae for the number of representations of a positive integer $n$ when $(a,b)=(1,1)$, $(1,3)$, $(2,3)$, $(1,9)$.

math.NT↗

arXiv2016

Elementary Evaluation of Convolution Sums involving primitive Dirichlet Characters for a Class of positive Integers

We extend the results obtained by E. Ntienjem to all positive integers. Let $\EuFrak{N}$ be the subset of $\mathbb{N}$ consisting of $\,2^ν\mho$, where $ν$ is in $\{0,1,2,3\}$ and $\mho$ is a squarefree finite product of distinct odd primes. We discuss the evaluation of the convolution sum, $\underset{\substack{ {(l,m)\in\mathbb{N}^{2}} {α\,l+β\,m=n} } }{\sum}σ(l)σ(m)$, when $αβ$ is in $\mathbb{N}\setminus\EuFrak{N}$. The evaluation of convolution sums belonging to this class is achieved by applying modular forms and primitive Dirichlet characters. In addition, we revisit the evaluation of the convolution sums for $αβ=9$, $16$, $18$, $25$, $36$. If $αβ\equiv 0 \pmod{4}$, we determine natural numbers $a,b$ and use the evaluated convolution sums together with other known convolution sums to carry out the number of representations of $n$ by the octonary quadratic forms $a\,(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})+ b\,(x_{5}^{2} + x_{6}^{2} + x_{7}^{2} + x_{8}^{2})$. Similarly, if $αβ\equiv 0 \pmod{3}$, we compute natural numbers $c,d$ and make use of the evaluated convolution sums together with other known convolution sums to determine the number of representations of $n$ by the octonary quadratic forms $c\,(\,x_{1}^{2} + x_{1}x_{2} + x_{2}^{2} + x_{3}^{2} + x_{3}x_{4} + x_{4}^{2}\,) + d\,(\,x_{5}^{2} + x_{5}x_{6} + x_{6}^{2} + x_{7}^{2} + x_{7}x_{8} + x_{8}^{2}\,)$. We illustrate our method with the explicit examples $αβ= 3^{2}\cdot 5$, $αβ= 2^{4}\cdot 3$, $αβ= 2\cdot 5^{2}$ and $αβ= 2^{6}$, .

math.NT↗

arXiv2016

The convolution sum $\sum_{al+bm=n} σ(l) σ(m)$ for $(a,b)=(1,28), (4,7), (1,14), (2,7), (1,7)$

We evaluate the convolution sum $\displaystyle W_{a,b}(n):= \sum_{al+bm=n} \hspace{-3mm} σ(l) σ(m)$ for $(a,b)=(1,28), (4,7), (2,7)$ for all positive integers $n$. We use a modular form approach. We also re-evaluate the known sums $W_{1,14}(n)$ and $W_{1,7}(n)$ with our method. We then use these evaluations to determine the number of representations of $n$ by the octonary quadratic form $x_1^2 + x_2^2 +x_3^2 + x_4^2 + 7(x_5^2 + x_6^2 + x_7^2 + x_8^2)$. Finally we compare our evaluations of the sums $W_{1,7}(n)$ and $W_{1,14}(n)$ with the evaluations of Lemire and Williams [10] and Royer [13] to express the modular forms $Δ_{4,7}(z)$, $Δ_{4,14, 1}(z)$ and $Δ_{4,14, 2}(z)$ (given in [10, 13]) as linear combinations of eta quotients.

math.NT↗

arXiv2016

Evaluation of the Convolution Sums $\underset{\substack{ {(l,m)\in\mathbb{N}_{0}^{2}} {α\,l+β\,m=n} } }{\sum}σ(l)σ(m)$, where $αβ=44,52$

The convolution sum, $\underset{\substack{ {(l,m)\in\mathbb{N}_{0}^{2}} {α\,l+β\,m=n} } }{\sum}σ(l)σ(m)$, where $αβ=44,52$, is evaluated for all natural numbers $n$. We then use these convolution sums to determine formulae for the number of representations of a natural number by the octonary quadratic forms $a\,(x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + x_{4}^{2})+ b\,(x_{5}^{2} + x_{6}^{2} + x_{7}^{2} + x_{8}^{2})$, where $(a,b)= (1,11),(1,13)$.

math.NT↗

arXiv2016

Evaluation of the Convolution Sum involving the Sum of Divisors Function for 14, 22 and 26

For all natural numbers $n$, we discuss the evaluation of the convolution sum, $\underset{\substack{{(l,m) \in \mathbb{N}_0^2} \\ {α\,l+β\,m=n} } }{\sum}σ(l)σ(m)$, where $αβ=14,22,26$. We generalize the extraction of the convolution sum using Eisenstein forms of weight $4$ for all pairs of positive integers $(α,β)$. We also determine formulae for the number of representations of a positive integer by the octonary quadratic forms $a\,(x_1^2 + x_2^2 + x_3^2 + x_4^2)+ b\,(x_5^2 + x_6^2 + x_7^2 + x_8^2)$, where $(a,b)= (1,1), (1,3), (2,3), (1,9)$. These numbers of representations of a positive integer are applications of the evaluation of certain convolution sums by J. G. Huard et al., A. Alaca et al. and D. Ye.

math.NT↗