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An alternative proof of the asymptotic formula for the Fourier coefficients of the elliptic modular $j$-function

Karin Ikeda

TL;DR

This paper studies the asymptotics of the Fourier coefficients $c_n$ of the elliptic j-function using a probabilistic framework inspired by Báez-Duarte. It provides two proofs: (i) a direct analysis based on a Jacobi-theta–decomposition of $j$ and a strong Gaussian condition that yields $c_n \sim \frac{e^{4\pi\sqrt{n}}}{\sqrt{2}\,n^{3/4}}$, and (ii) a Hauptmodul-based approach expressing $j$ as a rational function in Hauptmoduln $j_N$ and applying the same probabilistic method to eta-quotient expressions. A general lemma shows that certain eta-quotient products satisfy the strong Gaussian condition, enabling the asymptotics to be inferred from the probabilistic framework; explicit results are provided for several Hauptmodul cases and the method is indicated to extend to others with available eta-quotient representations. The results reinforce a probabilistic viewpoint on modular-form coefficient asymptotics and propose a broad, modular-analytic pathway for similar problems.

Abstract

In 1997, Báez-Duarte gave a probabilistic proof of the asymptotic formula for the partition function, which had originally been proved by Hardy-Ramanujan. Based on the probabilistic approach, this paper proves an asymptotic formula for the coefficients of the elliptic modular $j$-function using various expressions in terms of modular functions having simple infinite products.

An alternative proof of the asymptotic formula for the Fourier coefficients of the elliptic modular $j$-function

TL;DR

This paper studies the asymptotics of the Fourier coefficients of the elliptic j-function using a probabilistic framework inspired by Báez-Duarte. It provides two proofs: (i) a direct analysis based on a Jacobi-theta–decomposition of and a strong Gaussian condition that yields , and (ii) a Hauptmodul-based approach expressing as a rational function in Hauptmoduln and applying the same probabilistic method to eta-quotient expressions. A general lemma shows that certain eta-quotient products satisfy the strong Gaussian condition, enabling the asymptotics to be inferred from the probabilistic framework; explicit results are provided for several Hauptmodul cases and the method is indicated to extend to others with available eta-quotient representations. The results reinforce a probabilistic viewpoint on modular-form coefficient asymptotics and propose a broad, modular-analytic pathway for similar problems.

Abstract

In 1997, Báez-Duarte gave a probabilistic proof of the asymptotic formula for the partition function, which had originally been proved by Hardy-Ramanujan. Based on the probabilistic approach, this paper proves an asymptotic formula for the coefficients of the elliptic modular -function using various expressions in terms of modular functions having simple infinite products.

Paper Structure

This paper contains 4 sections, 7 theorems, 118 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries for Probabilistic Methods
  3. Proof of the asymptotic formula \ref{['jasy']}
  4. Another proof via Hauptmodul

Key Result

Lemma 2.2

Let $F(t)$ be a power series satisfying the strong Gaussian condition. Suppose $\widetilde{m}(t)$ and $\widetilde{\sigma}(t)$ are continuous functions such that $\widetilde{m}(t)\to \infty\ (t\to R)$ monotonically, $\widetilde{\sigma}(t)/\sigma(t)\to 1$ as $t\to R$, and Then, as $n\to \infty$, where $\tau_n$ is the unique solution of the equation $\widetilde{m}(\tau_n)=n$ for large enough $n$.

Theorems & Definitions (16)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • Theorem 2.4
  • Lemma 3.1: Kaneko
  • proof
  • Theorem 3.2
  • Definition 3.3
  • Theorem 3.4
  • ...and 6 more