On a level analog of Selberg's result on $S(t)$

1. Problem Statement

The paper addresses the statistical distribution of the argument of $L$-functions associated with holomorphic Hecke cusp forms in the level aspect.

• Context: Selberg's classical work established that for the Riemann zeta function $\zeta(s)$ and Dirichlet $L$-functions $L(s, \chi)$, the argument $S(T) = \frac{1}{\pi} \arg L(1/2 + iT)$ behaves like a random variable with a Gaussian distribution when normalized by $\sqrt{\log \log T}$ (or $\sqrt{\log \log q}$ for Dirichlet characters). Specifically, Selberg proved asymptotic formulas for the moments of $S(T)$.
• Gap: While analogues of these results exist for higher-rank $L$-functions in the spectral (eigenvalue) and weight aspects, often relying on the Generalized Riemann Hypothesis (GRH), an unconditional result for $GL_2$ $L$-functions in the level aspect (varying the level $q$ of the cusp form) was missing.
• Objective: The authors aim to establish an unconditional asymptotic formula for the moments of $S(t, f) = \frac{1}{\pi} \arg L(1/2 + it, f)$, where $f$ is a holomorphic Hecke cusp form of weight 2 and prime level $q$. They further aim to derive a central limit theorem for the distribution of these arguments as $q \to \infty$.

2. Methodology

The authors employ a combination of analytic number theory techniques, specifically adapting Selberg's moment method to the family of cusp forms with varying levels.

A. Approximation via Truncated Dirichlet Series

The core of the proof involves approximating the argument function $S(t, f)$ by a truncated Dirichlet series.

• Definition: They define $S(t, f)$ via continuous variation of the argument of $L(s, f)$ along the line $\text{Im}(s) = t$.
• Approximation: Using the logarithmic derivative $L'/L$ and a smooth cutoff function $\Lambda_x(n)$, they derive an approximation (Theorem 3.4):
  $$S(t, f) \approx \frac{1}{\pi} \text{Im} \sum_{n \le x^3} \frac{C_f(n) \Lambda_x(n)}{n^{1/2+it} \log n}$$
  where $C_f(n)$ are coefficients related to the Satake parameters of $f$.
• Error Control: The error term depends on the distance of the zeros of $L(s, f)$ from the critical line. To handle this without assuming GRH, they introduce a parameter $\sigma_x$ that dynamically shifts based on the location of the zeros (specifically, $\sigma_x = 1/2 + \max(|\beta - 1/2|, 5/\log x)$).

B. Weighted Moments and Trace Formula

To compute the moments over the family of forms, they utilize the Petersson Trace Formula.

• Harmonic Weights: Instead of a simple sum, they use harmonic weights $\omega_f = \frac{1}{4\pi \langle f, f \rangle}$, which satisfy $\sum \omega_f = 1 + o(1)$.
• Orthogonality: The trace formula allows them to evaluate sums of products of Fourier coefficients $\sum_f \omega_f \lambda_f(m) \lambda_f(n)$. This reduces the problem to estimating diagonal terms ($m=n$) and bounding off-diagonal terms using bounds on Kloosterman sums and Bessel functions.

C. Zero-Density Estimates

A critical component is controlling the "bad" zeros (those far from the critical line) that could spoil the approximation.

• Tool: They employ a weighted zero-density estimate (Lemma 2.4) established by Kowalski and Michel. This estimate bounds the number of zeros $\rho = \beta + i\gamma$ with $\beta > \sigma$ in the family, weighted by $\omega_f$.
• Application: This allows them to prove that the contribution of forms with zeros far from the critical line is negligible, ensuring the approximation holds for "almost all" forms in the family without assuming GRH.

D. Moment Calculation

• They expand the $n$-th power of the main term $M(t, f)$ (the truncated sum).
• Using the multiplicative properties of Hecke eigenvalues and the orthogonality relations, they show that only specific terms (where exponents match in pairs) contribute significantly to the sum.
• This leads to a combinatorial calculation involving factorials and powers of $\log \log q$.

3. Key Contributions and Results

Main Theorem (Theorem 1.1): Asymptotic Moments

For a fixed $t > 0$ and integer $n$, as the prime level $q \to \infty$:
$$\sum_{f \in H_2(q)} \omega_f \cdot S(t, f)^n = C_n (\log \log q)^{n/2} + O_{t,n}((\log \log q)^{(n-1)/2})$$
where $C_n$ is the $n$-th moment of a Gaussian distribution:
$$C_n = \begin{cases} \frac{n!}{(n/2)! (2\pi)^n} & \text{if } n \text{ is even} \\ 0 & \text{if } n \text{ is odd} \end{cases}$$
Significance: This confirms that the magnitude of $S(t, f)$ is on average of order $\sqrt{\log \log q}$, mirroring Selberg's results for $\zeta(s)$ and Dirichlet $L$-functions, but achieved unconditionally (without GRH) in the level aspect.

Central Limit Theorem (Theorem 1.2)

The authors define a probability measure $\mu_q$ on $\mathbb{R}$ based on the normalized arguments:
$$\mu_q(E) = \frac{\sum_{f} \omega_f \cdot \mathbf{1}_E\left( \frac{S(t, f)}{\sqrt{\log \log q}} \right)}{\sum_{f} \omega_f}$$
They prove that as $q \to \infty$, $\mu_q$ converges weakly to a Gaussian distribution with mean 0 and variance $(2\pi^2)^{-1}$:
$$\lim_{q \to \infty} \mu_q([a, b]) = \sqrt{\pi} \int_a^b e^{-\pi^2 \xi^2} d\xi$$

4. Significance

1. Unconditional Result: Previous work on higher-rank or level-aspect analogues often relied on the Generalized Riemann Hypothesis (GRH). This paper removes that dependency, providing a rigorous unconditional proof for $GL_2$ in the level aspect.
2. Completing the Analogy: It fills a gap in the "Selberg's limit theorem" landscape, showing that the Gaussian behavior of the argument of $L$-functions is a robust phenomenon across different families (Riemann zeta, Dirichlet characters, and holomorphic cusp forms) and different aspects (t, weight, and level).
3. Methodological Advancement: The paper successfully adapts the "truncated Dirichlet series" approximation technique to the level aspect, overcoming the technical difficulty of varying levels by combining it with weighted zero-density estimates. This methodology could be applicable to other families of $L$-functions.
4. Statistical Insight: The result provides deep insight into the distribution of zeros of $L$-functions, suggesting that the fluctuations of the argument are governed by random matrix theory statistics (specifically the Gaussian Unitary Ensemble behavior implied by the Gaussian limit).

In summary, Sun and Wang have successfully extended Selberg's classical probabilistic results on the argument of $L$-functions to the family of holomorphic cusp forms of prime level, proving that the distribution is Gaussian and establishing the precise asymptotic moments without assuming unproven hypotheses.