Convolution identities for divisor sums and modular forms

• 成果类型: Article
• 署名作者: Fedosova, Ksenia; Klinger-Logan, Kim; Radchenko, Danylo
• 署名单位: University of Munster; Kansas State University; Universite de Lille
• 刊物名称: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• ISSN/ISSBN: 0027-9719
• DOI: 10.1073/pnas.2322320121
• 发表日期: 2024-10-29
• 关键词: eisenstein series

摘要

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 X n1 is an element of ZN{0,n} cP ( n 1 , n - n 1 ) 6 2 m 1 ( n 1 ) 6 2 m 2 ( n - n 1 ), where tp ( n 1 , n 2 ) is a Laurent polynomial with logarithms. Contrary to original expectations, such convolution sums, suitably extended to n 1 is an element of { 0 , n }, do not vanish, but instead, they carry number theoretic meaning in the form of Fourier coefficients of holomorphic cusp forms.