The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

Abstract

We study the algebra MD of generating function for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in Q arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra MD is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in MD. The (quasi-)modular forms for the full modular group Sl_2(Z) constitute a sub-algebra of MD this also yields linear relations in MD. Generating functions of multiple divisor sums can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those in length 2, coming from modular forms.

Some new results added, notation slightly changed and minor mistakes were corrected

References in corpus (1)

• The Algebra of a q-Analogue of Multiple Harmonic Series

Cited by in corpus (5)

• Multiple $q$-zeta brackets
• Diagrammatic Expansion of Non-Perturbative Little String Free Energies
• The double shuffle relations for multiple Eisenstein series
• The algebra of bi-brackets and regularised multiple Eisenstein series
• A short note on a conjecture of Okounkov about a q-analogue of multiple zeta values