[Paper Review] Rademacher Sums and Rademacher Series

Miranda C. N. Cheng, John F. R. Duncan|arXiv (Cornell University)|Oct 10, 2012

Advanced Mathematical Identities41 references4 citations

TL;DR

This paper introduces Rademacher sums and Rademacher series as tools for constructing modular and mock modular forms, especially in non-convergent cases. It establishes a general regularization method for Poincaré series, derives explicit Fourier coefficient formulas via Rademacher series, and reveals a universal Zagier duality between coefficients of dual-weight mock modular forms, with applications to monstrous and umbral moonshine and physical interpretations in string theory.

ABSTRACT

We exposit the construction of Rademacher sums in arbitrary weights and describe their relationship to mock modular forms. We introduce the notion of Rademacher series and describe several applications, including the determination of coefficients of Rademacher sums and a very general form of Zagier duality. We then review the application of Rademacher sums and series to moonshine both monstrous and umbral and highlight several open problems. We conclude with a discussion of the interpretation of Rademacher sums in physics.

Motivation & Objective

• To develop a regularization procedure for Poincaré series in low weights where standard convergence fails.
• To define Rademacher series as a two-dimensional grid of values that encode Fourier coefficients of modular and mock modular forms.
• To establish a general form of Zagier duality between coefficients of mock modular forms in dual weights.
• To apply Rademacher sums and series to the study of monstrous and umbral moonshine.
• To explore physical interpretations of Rademacher sums in string theory and conformal field theory.

Proposed method

• Regularization of non-absolutely convergent Poincaré series using a limiting sum over coset representatives with bounded parameters, inspired by Rademacher's work on the j-invariant.
• Construction of Rademacher sums as regularized sums over cosets of the stabilizer subgroup $̳_{\infty}$ in $\mathrm{SL}_2(\mathbb{Z})$, with weight $w=2k$ and exponential input $f(\tau) = e(m\tau)$.
• Introduction of Rademacher series as a two-dimensional array of values derived from the coefficients of Rademacher sums, with connections to Eichler integrals of modular forms.
• Use of special functions such as the incomplete gamma function $\gamma(s,x)$, the modified Bessel function $I_\alpha(z)$, and the Dedekind eta function $\eta(\tau)$ in the analytic framework.
• Application of the Lipschitz summation formula and its extension to $s=1$ to handle conditional convergence in low-weight cases.
• Employment of Appell-Lerch sums $\mu(\tau,z)$ and Jacobi theta functions to describe transformation properties and modular behavior.

Experimental results

Research questions

• RQ1 How can Poincaré series be regularized in weights where standard summation fails to converge?
• RQ2 What is the structure of the Fourier coefficients of Rademacher sums, and how can they be expressed in series form?
• RQ3 How do Rademacher series encode information about Eichler integrals and dual-weight mock modular forms?
• RQ4 What is the nature of the generalized Zagier duality revealed by Rademacher series?
• RQ5 How do Rademacher sums contribute to the construction and understanding of moonshine phenomena, including monstrous and umbral moonshine?

Key findings

• The paper establishes that Rademacher sums provide a regularization of Poincaré series in weight $w=0$, extending Rademacher's original construction of the $j$-invariant as a conditionally convergent sum.
• For weights $w \leq 2$, Rademacher sums converge conditionally and define mock modular forms, with the weight $w$ action twisted by a modular form of weight $2-w$.
• The coefficients of Rademacher sums are expressed via Rademacher series, which form a two-dimensional grid of values, with half of them corresponding to the coefficients of the sum and the other half to Eichler integrals of modular forms.
• A universal form of Zagier duality is proven: the coefficients of two families of mock modular forms in dual weights coincide up to sign, as encoded in the Rademacher series.
• The construction applies to monstrous moonshine, where Rademacher sums recover the graded traces of the monster group, and to umbral moonshine, providing a framework for the Mathieu group case.
• Physical interpretations are suggested, linking Rademacher sums to partition functions and black hole entropy in string theory, particularly through their modular properties and connection to BPS states.