Convolution identities for divisor sums and modular forms

2023/12/01 by Ksenia Fedosova, Fedosova, Ksenia, Kim Klinger-Logan +3 · 1 citation
Mathematics · #Analytic Number Theory Research #Algebraic Geometry and Number Theory #Advanced Mathematical Identities

paper · pdf · doi:10.48550/arxiv.2312.00722

Abstract

We prove exact identities for convolution sums of divisor functions of the form ∑_n₁ ∈ ℤ \smallsetminus \0,n\φ(n₁,n-n₁)σ_2m1(n₁)σ_2m2(n-n₁) where φ(n₁,n₂) is a Laurent polynomial with logarithms for which the sum is absolutely convergent. Such identities are motivated by computations in string theory and prove and generalize a conjecture of Chester, Green, Pufu, Wang, and Wen from \citeCGPWW. Originally, it was suspected that such sums, suitably extended to n₁∈\0,n\ should vanish, but in this paper we find that in general they give Fourier coefficients of holomorphic cusp forms.

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• Convolution identities for complex-indexed divisor functions and modular graph functions

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