Evaluation of the convolution sums ∑l+27m=nσ(l)σ(m) and ∑l+32m=nσ(l)σ(m)

Abstract

We determine the convolution sums [Formula: see text] and [Formula: see text] for all positive integers [Formula: see text]. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of [Formula: see text] by the octonary quadratic forms [Formula: see text] and [Formula: see text]. A modular form approach is used.

On the number of representations of an integer by certain quadratic forms in sixteen variables

“…Then Let N 1 (n) be the number of representations of the positive integer n by the quadratic form Q 16 = Q 8 ⊕ Q 8 , where Q 8 is defined by (3). Note that Q 16 is the quadratic form x 2 1 + x 1 x 2 + x 2 2 + x 2 3 + x 3 x 4 + x 2 4 + x 2 5 + x 5 x 6 + x 2 6 + x 2 7 + x 7 x 8 + x 2 8 + x 2 9 + x 9 x 10 + x 2 10 + x 2 11 + x 11 x 12 + x 2 …”

“…The convolution sums W 1,1 1,2 (n), W 1,1 1,4 (n), W 1,3 1,2 (n), W 1,3 2,1 (n), W 1,1 1,5 (n), W 1, 2 1,5 (n) and W 2,1 1,5 (n) have been computed by Royer [19] and the convolution sums W 1, 3 1,3 (n) and W 3,1 1,3 (n) have been evaluated recently by Yao and Xia [24]. Convolution sums involving the divisor function σ(n) have been extensively evaluated by Williams and his coauthors (see, for example, ( [1-8, 10, 11, 14, 22, 23]).…”

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

“…Proof. These identities follow immediately on taking (α, β) = (1, 33), (3,11), (1,40), (5,8), (1, 56), (7,8) in Lemma 3.3. In case αβ = 40 we take all n in {1, 2, .…”

“…Alaca (1,11), (1,13) E. Ntienjem [24] (1,12), (1,16), (3,4) E. Ntienjem [23] K. S. Williams [30] (1,4), (1,6), (1,8), ( (1,9) Ş . Alaca & Y. Kesicioǧlu [5] (1,10), (2,5) E. Ntienjem [22] (1,12), (3,4) D. Ye [34] (1,16) E. Ntienjem [23] Table 3: Known representations of n by the form Equation 1.5…”

“…One can then make use of Equation 6.14, Equation 5.12 and Equation 5.13 to simplify this formula.7.1.2. Representations by Octonary Quadratic Forms Equation 1.5.We give formulae for the number of representations of a positive integer n by the Octonary Quadratic Form Equation 1.4 by mainly applying the evaluation of the convolution sums W (1,40) (n), W (1,56) (n), W(5,8) (n) and W(7,8) (n). To achieve that, we recall that 40 = 2 3 · 5 and 56 = 2 3 · 7 are of the restricted form in Section 4.1.…”