Limiting Distribution and Rate of Convergence for GL(3) Fourier Coefficients

Technical Summary of "Limiting Distribution and Rate of Convergence for GL(3) Fourier Coefficients"

Problem Statement
This paper addresses the probabilistic behavior of the error term associated with the summatory function of Fourier coefficients of automorphic forms on $GL(3)$. Specifically, let $f$ be a fixed self-dual Hecke–Maass cusp form for $GL(3)$ with normalized Fourier coefficients $A_f(n, m)$. The author defines the error term $\Delta_f(x) = \sum_{n \le x} A_f(n, 1)$. The central problem is to determine whether the normalized function $F_f(t) = t^{-1/3}\Delta_f(t)$ possesses a limiting distribution as $t \to \infty$, and to quantify the rate of convergence of this distribution.

This work serves as an analogue to a seminal result by Heath-Brown (1992) regarding the classical divisor problem and the triple divisor function $\Delta_3(x)$. While Heath-Brown established the existence of a limiting distribution for the normalized error term in the divisor problem, this paper extends that framework to the setting of $GL(3)$ automorphic forms.

Methodology
The author adapts the probabilistic approximation methods introduced by Heath-Brown and refined by subsequent works (e.g., Lamzouri, Lau). The core strategy involves approximating the deterministic error term $F_f(t)$ by a random trigonometric series constructed from independent random variables.

Key methodological components include:

1. Random Model Construction: A random variable $F_{f,X}$ is defined as a random trigonometric series involving the Fourier coefficients $A_f(1, n)$ and independent random variables $\{X_n\}$ uniformly distributed on $[0, 1]$. The series is constructed to mimic the oscillatory nature of $\Delta_f(t)$ derived from Voronoï-type summation formulas.
2. Mean Value Estimates vs. Pointwise Estimates: A critical distinction in this work is the reliance on mean value estimates of Fourier coefficients rather than pointwise bounds. The author explicitly notes that strong pointwise estimates for $GL(3)$ coefficients (such as the Ramanujan conjecture) are not assumed. Instead, the proof utilizes the mean square estimate of partial sums of Fourier coefficients.
3. Analytic Tools:
   • L-function Bounds: The proof of the mean square estimate (Theorem 1.2) relies on establishing a bound for the second moment of the $L$-function $L(s, f)$ on the critical line. Specifically, the author utilizes a result by Pal [22] to show that $\int_1^T |L(\sigma + it, f)|^2 dt \ll T^{1+\varepsilon}$ for a specific $\sigma < 2/3$ (specifically $\sigma = 13/20$).
   • Moment Matching: To prove the convergence of distributions, the paper demonstrates that the moments of the truncated deterministic function $F_f(t)$ closely approximate the moments of the random model $F_{f,X}$ over a specific range. This involves bounding off-diagonal terms using the linear independence of cube roots over $\mathbb{Q}$.
   • Laplace Transform Approximation: The distribution of large values is analyzed by approximating the Laplace transform of $F_f(t)$ with that of $F_{f,X}$, utilizing bounds on the exponential moments of the random series.

Key Contributions and Results

• Theorem 1.1 (Existence of Limiting Distribution): The paper proves that $F_f(t) = t^{-1/3}\Delta_f(t)$ possesses a limiting distribution function $g(\alpha)$. The density function $g(\alpha)$ is shown to be smooth ($C^\infty$), rapidly decaying (Schwartz class), and extendable to an entire function on $\mathbb{C}$. This result holds without assuming the Ramanujan conjecture.

• Theorem 1.2 (Mean Square Estimate): A quantitative estimate for the mean square of the error term is established:
  $$\int_0^x \Delta_f(y)^2 dy = \frac{1}{10\pi^2} \left( \sum_{n=1}^\infty \frac{A_f(1, n)^2}{n^{4/3}} \right) x^{5/3} + O(x^{18/11+\varepsilon}).$$
  This result is derived by finding a $\sigma < 2/3$ such that the second moment of the $L$-function is bounded by $T^{1+\varepsilon}$, a step made possible by recent work of Pal.

• Theorem 1.3 (Rate of Convergence): The paper provides a quantitative bound on the discrepancy between the distribution function of $F_f(t)$ and the random model $F_{f,X}$. Specifically, the discrepancy $D_F(T)$ satisfies:
  $$D_F(T) \ll \frac{(\log \log \log T)^{5/3}}{(\log \log T)^{1/3}}.$$
  This confirms that the probabilistic model accurately describes the statistical behavior of the error term with a specific rate of convergence.

• Theorem 1.4 (Distribution of Large Values): The paper establishes upper and lower bounds for the probability that $F_f(t)$ exceeds a large value $V$. For $V$ in a specific range involving iterated logarithms, the probability decays exponentially:
  $$\exp(-b_3 V^{35/2+\varepsilon}) \le P_T(F_f(t) > V) \le \exp(-b_4 V^{5/3-\varepsilon}).$$
  The author notes that due to the lack of strong pointwise estimates for $GL(3)$ coefficients, the exponents in these bounds are weaker than those known for the classical divisor problem.

Significance and Claims
The paper claims to successfully extend the probabilistic theory of error terms from the classical divisor problem to the more complex setting of $GL(3)$ automorphic forms. Its primary significance lies in:

1. Generalization: Demonstrating that the limiting distribution phenomenon is not unique to divisor functions but applies to Fourier coefficients of higher-rank automorphic forms.
2. Methodological Adaptation: Showing that the probabilistic approximation method can be adapted to settings where pointwise bounds are unavailable, by substituting them with mean value estimates and $L$-function moment bounds.
3. Quantitative Precision: Providing explicit rates of convergence and bounds for large deviations, which are essential for understanding the fine-scale distribution of these arithmetic functions.

The author explicitly states that the requirement for $f$ to be self-dual is necessary to ensure the convergence of the defining series for the random model. The paper concludes that if convergence can be proven for non-self-dual forms, the results would extend to those cases as well. The work does not propose new experimental applications but rather deepens the theoretical understanding of the distribution of arithmetic error terms in the context of the Langlands program.