2025/12/24 by Ksenia Fedosova, Fedosova, Ksenia, Kim Klinger-Logan +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2512.21413
openalex publication_date 2025/12/24 · openalex created_date 2025/12/30 · openalex updated_date 2026/07/28
We find exact identities for sums of the form ∑_\stackreln1+n2 = nn1 ∈ ℤ ∖ \ 0, n \ Q(n1,n2) σ-r1(n1) σ-r2(n2), where n∈ℕ, r1,r2∈ℂ, Q is a combination of hypergeometric functions, and σa(x) denotes the divisor function. Specifically, we find that they can be expressed in terms of Fourier coefficients of Hecke cusp forms weighted by their L-values. This result expands upon previous work with Radchenko in which such identities were found for divisor functions with even integer index \citeFKLR and encompasses results of Jacobi \citemotohashi1994binary and Diamantis and O'Sullivan in \citediamantis2010kernels, o2023identities for divisor functions with odd integer index. The proof of our result expresses these sums in terms of Estermann zeta functions and uses trace formulae. In addition, we use a regularization of divergent convolution sums to provide a mathematical explanation for L-values (non-critical in the sense of Deligne) appearing in modular graph functions \citeDKS20212.