Evaluation of the convolution sums W_1,42(n), W_2,21(n), W_3,14(n) and W_6,7(n)

2022/06/30 by Köklüce, Bülent
#11E20 #11E25 #11F11 #11F20 #11F27 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2207.09516

Abstract

In this paper, we use a modular form approach to evaluate the convolution sums ∑_l+42m=nσ(l)σ(m), ∑_2l+21m=nσ(l)σ(m), ∑_3l+14m=nσ(l)σ(m) and ∑_6l+7m=nσ(l)σ(m) for all positive integers n, and then use their evaluations to determine the number of representation of a positive integer n by the quadratic form x₁² +x₁x₂ +x₂² +x₃² +x₃x₄ +x₄² + 14(x₅² +x₅x₆ +x₆² +x₇² +x₇x₈ +x₈²).

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