A Weighted Sum Formula for Double Eisenstein Series

Henrik Bachmann

arXiv:2607.21358v1 Primary: math.NT Secondary: math.CO

Abstract

We prove a weighted sum formula for double Eisenstein series. Its corresponding identity for the generating series of multiple divisor sums was conjectured by the author in his master's thesis. The double Eisenstein series identity follows from the restricted double-shuffle relations proved by the author and Tasaka, while the proof of the divisor-sum identity is combinatorial and uses generating series.

Disclosure

“0 Equating (3.5) and (3.6), substituting the definitions of Ar , Ar,s , gives (1.4) by (2.2). □ AI & computational resource disclosure: The main results, their formulation, and the underlying idea of proof are the author’s own. ChatGPT 5.6 and Claude Fable 5 were used throughout as research assistants: they carried out and checked computations, and a number of intermediate steps in the proofs were worked out in dialogue with these systems. All statements and proofs were”
PDF page 6

Classification: Proof ideas or individual proof-step assistance
Multiplier: 8
Verified: 2026-08-23T19:05:17.624957Z

Structural counts

Pages 6 pdf

Theorems 2 source

Lemmas 1 source

Propositions 0 source

Corollaries 0 source

Definitions 0 source

Displayed equations 42 source

Bibliography entries 5 source

Appendix pages 0 estimated

Count notes

Source counts use the expanded primary TeX file Bachmann_DESWeightedSum.tex.
Appendix pages include the first PDF page with an explicit Appendix heading through the final page.