Explicit evaluation of triple convolution sums of the divisor functions

Article Type

Research Article

Publication Title

International Journal of Number Theory

Abstract

In this paper, we use the theory of modular forms and give a general method to obtain the convolution sums (Formula presented) for odd integers r1, r2, r3 ≥ 1, and d1, d2, d3, n ∈ N, where σr(n) is the sum of the rth powers of the positive divisors of n. We consider four cases, namely (i) r1 = r2 = r3 = 1, (ii) r1 = r2 = 1; r3 ≥ 3 (iii) r1 = 1; r2, r3 ≥ 3 and (iv) r1, r2, r3 ≥ 3, and give explicit expressions for the respective convolution sums. We provide several examples of these convolution sums in each case and further use these formulas to obtain explicit formulas for the number of representations of a positive integer n by certain positive definite quadratic forms. The existing formulas for W1,1,1(n) (in [20]), W1,1,2(n), W1,2,2(n), W1,2,4(n) (in [7]), W1,1,11,3,3(n), W1,1,31,3,3(n), W1,3,31,3,3(n), W3,1,11,3,3(n), W3,3,11,3,3(n) (in [35]), Wd1,d2,d3(n), lcm(d1, d2, d3) ≤ 6 (in [30]) and lcm(d1, d2, d3) = 7, 8, 9 (in [31]), which were all obtained by using the theory of quasimodular forms, follow from our method.

First Page

1073

Last Page

1098

DOI

10.1142/S1793042124500544

Publication Date

5-1-2024

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