Interpolated sequences and critical $L$-values of modular forms

TL;DR

This paper explores the connection between interpolated sequences related to special values of the Riemann zeta function and critical L-values of modular forms, extending known results to new families of sequences and modular forms.

Contribution

It generalizes Zagier's results by linking interpolated sequences to critical L-values of modular forms of various weights, including an infinite family of such evaluations.

Findings

Interpolations of Zagier's six sequences relate to weight 3 modular forms.

Established an infinite family connecting Brown's integrals to odd weight modular forms.

Extended the understanding of the relationship between special sequences and modular form L-values.

Abstract

Recently, Zagier expressed an interpolated version of the Ap\'ery numbers for $\zeta(3)$ in terms of a critical $L$-value of a modular form of weight 4. We extend this evaluation in two directions. We first prove that interpolations of Zagier's six sporadic sequences are essentially critical $L$-values of modular forms of weight 3. We then establish an infinite family of evaluations between interpolations of leading coefficients of Brown's cellular integrals and critical $L$-values of modular forms of odd weight.