Evaluation of the convolution sums ∑_l+15m=n σ(l) σ(m) and ∑_3l+5m=n σ(l) σ(m) and some applications

2012/07/21 by B. Ramakrishnan, Ramakrishnan, B., Brundaban Sahu +1
Mathematics · #11E25 #11F11 #11F20 #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 11A25 #Secondary 11E20 #math.NT #msc:11A25 #msc:11E20 #msc:11E25 #msc:11F11 #msc:11F20

paper · pdf · doi:10.48550/arxiv.1207.5107

To appear in IJNT

openalex publication_date 2012/07/21 · arxiv created 2012/10/21 · arxiv updated 2012/10/23 · openalex created_date 2024/04/10 · openalex updated_date 2026/07/28

Abstract

We evaluate the convolution sums ∑_l,m∈ \mathbb N, l+15m=n σ(l) σ(m) and ∑_l,m∈ \mathbb N, 3l+5m=n σ(l) σ(m) for all n∈ \mathbb N using the theory of quasimodular forms and use these convolution sums to determine the number of representations of a positive integer n by the form x₁² + x₁x₂ + x₂² + x₃² + x₃x₄ + x₄² + 5 (x₅² + x₅x₆ + x₆² + x₇² + x₇x₈ + x₈²). We also determine the number of representations of positive integers by the quadratic form x₁² + x₂²+x₃²+x₄² + 6 (x₅²+x₆²+x₇²+x₈²), by using the convolution sums obtained earlier by Alaca, Alaca and Williams \citeaw3, aw4.

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