Convolution identities for divisor sums and modular forms

We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums ∑ n 1 ∈ Z ∖ { 0 , n } φ ( n 1 , n − n 1 ) σ 2 m 1 ( n 1 ) σ 2 m 2 ( n −...

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Published inProceedings of the National Academy of Sciences - PNAS Vol. 121; no. 44; p. e2322320121
Main Authors Fedosova, Ksenia, Klinger-Logan, Kim, Radchenko, Danylo
Format Journal Article
LanguageEnglish
Published United States National Academy of Sciences 29.10.2024
Subjects
Convolution
Physical Sciences
Polynomials
String theory
modular forms
L-values
graviton scattering
convolution sums
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ISSN0027-8424
1091-6490
1091-6490
DOI10.1073/pnas.2322320121

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Summary:We consider certain convolution sums that are the subject of a conjecture by Chester, Green, Pufu, Wang, and Wen in string theory. We prove a generalized form of their conjecture, explicitly evaluating absolutely convergent sums ∑ n 1 ∈ Z ∖ { 0 , n } φ ( n 1 , n − n 1 ) σ 2 m 1 ( n 1 ) σ 2 m 2 ( n − n 1 ) , where φ ( n 1 , n 2 ) is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to n 1 ∈ { 0 , n } , do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms.
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Edited by Kenneth Ribet, University of California, Berkeley, CA; received December 21, 2023; accepted August 24, 2024
1K.F., K.K.-L., and D.R. contributed equally to this work.
ISSN:0027-8424
1091-6490
1091-6490
DOI:10.1073/pnas.2322320121