Evaluation of Some Convolution Sums by Quasimodular Forms

Barış Kendirli

Abstract

We evaluate the convolution sums ∑_{l+135m=n}σ(l)σ(m),  ∑_{3l+45m=n}σ(l)σ(m),  ∑_{5l+27m=n}σ(l)σ(m),∑_{9l+15m=n}σ(l)σ(m), ∑_{l+45m=n}σ(l)σ(m),∑_{5l+9m=n}σ(l)σ(m), and ∑_{3l+15m=n}σ(l)σ(m) for all nεN using the theory of quasimodular forms and use some of these convolution sums to determine the number of representations of a positive integer n by some direct sum of the forms x₁²+x₁x₂+34x₂²,5x₁²+5x₁x₂+8x₂²,4x₁²±3x₁x₂+9x₂², 2x₁²±x₁x₂+17x₂² of discriminant -135

Keywords

Quasimodular forms, divisor functions, convolution sums, representation number 11A25,11F11,11F25,11F20

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