Evaluation of the Convolution Sums \underset\substack (l,m)∈ℕ₀² α l+β m=n ∑σ(l)σ(m), where αβ=44,52

2016/06/16 by Ebénézer Ntienjem, Ntienjem, Ebénézer
Mathematics · #Mathematical Approximation and Integration #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.1606.05155

Abstract

The convolution sum, \underset\substack (l,m)∈ℕ₀² α l+β m=n ∑σ(l)σ(m), where αβ=44,52, is evaluated for all natural numbers n. We then use these convolution sums to determine formulae for the number of representations of a natural number by the octonary quadratic forms a (x₁² + x₂² + x₃² + x₄²)+ b (x₅² + x₆² + x₇² + x₈²), where (a,b)= (1,11),(1,13).

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