On the efficient computation of Fourier coefficients of eta-quotients

TL;DR

The work studies Fourier coefficients of negative-weight eta-quotients via HRR-type series and establishes that the HRR coefficients $A_k(n)$ can be expressed as twisted Kloosterman sums, enabling efficient computation. It generalizes Lehmer’s prime-power multiplicativity to general $k$ and provides an explicit error bound for truncating the HRR series, making exact computation feasible. An algorithm is developed combining multiplicativity, Kloosterman sums, and explicit bounds, with examples demonstrating substantial speedups over direct summation. These results enhance practical computation of eta-quotient coefficients and illuminate their arithmetic structure through twisted sums and Dedekind-sum identities.

Abstract

The Fourier coefficients of a negative weight eta-quotient, in many particular cases, and after Sussman in general, are known to be expressible by Hardy-Ramanujan-Rademacher type series. We show that the central terms of the coefficients of these series can be efficiently computed, showing that they can be expressed in terms of twisted Kloosterman sums, and that they satisfy multiplicativity relations; this extends the results from Lehmer for the partition function. We also give explicit bounds for the tails of these series, needed for effectively computing the aforementioned Fourier coefficients.

Theorems & Definitions (50)

• Theorem : Theorem \ref{['thm:multiplicativity']}
• Theorem : Theorem \ref{['thm:kloosterman']}
• Remark 2.1
• Theorem 2.2: Sussman, 2017
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3