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

Abstract

.l/ .m/, where˛ˇD 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˛ˇD 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 .x 2

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

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We extend the results obtained by E. Ntienjem [23] to all positive integers. Let N be the subset of N consisting of 2 ν 0, where ν is in {0, 1, 2, 3} and 0 is a squarefree finite product of distinct odd primes.

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“…p j , where gcd(c, d) = 1, 0 ≤ ν ≤ 3, E. Ntienjem [23] κ ∈ N, p j > 3 distinct primes Table 3: Known representations of n by the form Equation 1.4…”

“…Given α, β ∈ N such that α ≤ β, let the convolution sum be defined by Equation 1.2. E. Ntienjem [22,23] has shown and A. Alaca et al…”

“…(a, b) Authors References (1,1),(1,3), (1,9), (2,3) E. Ntienjem [23] (1,2) K. S. Williams [31] (1,4)…”

Representations of an Integer by Some Quaternary and Octonary Quadratic Forms

“…We now use the convolution sums W a,b (n) and W N (n) obtained by several authors (see the [2,5,16,18] (3, 2; 1) W N (n), N = 1, 2, 22 W 2,11 (n) [15,18] To get the formula for R 3,2;1 (n) we also need the convolution sum W 11 (n). Though this convolution sum is obtained by E. Royer in [18], it involved a pair of terms with complex coefficients.…”