An omega result for the first sign change of coefficients of symmetric power $L$-functions of Hecke-Maass cusp forms

Technical Summary: An Omega Result for the First Sign Change of Coefficients of Symmetric Power L-Functions of Hecke-Maass Cusp Forms

Problem Statement
The paper addresses the problem of determining the magnitude of the first sign change, denoted as $n_f$ (or $n_{\text{sym}^m u_k}$ in this context), for the coefficients of $L$-functions associated with automorphic forms. Specifically, it investigates the least positive integer $n$ such that the coefficient $\lambda(n)$ becomes negative.

While the behavior of the first sign change for holomorphic Hecke cusp forms has been studied extensively (notably by Lamzouri [6]), the analogous problem for Hecke-Maass cusp forms presents significant technical challenges. The primary difficulty lies in the absence of the Generalized Ramanujan Conjecture (GRC) for Maass forms. Without GRC, the Satake parameters $\alpha_{u_k, j}(p)$ are not guaranteed to lie on the unit circle, potentially leading to "exceptional" eigenvalues that grow exponentially. The paper aims to extend Lamzouri's omega results to Hecke-Maass forms and further generalize these results to the coefficients of symmetric power $L$-functions, $L(s, \text{sym}^m u_k)$, without assuming GRC.

Methodology
The authors employ a probabilistic sieve method adapted from Lamzouri's work, refined to handle the potential failure of the GRC. The core methodology involves the following steps:

1. Trigonometric Polynomials and Chebyshev Expansions:
   The authors utilize a specific trigonometric polynomial $f(\theta)$ constructed from Chebyshev polynomials of the first kind ($T_L$) and second kind ($X_\ell$). They define a non-negative function $g(\theta)$ which acts as a sieve weight. This function is expanded as $g(\theta) = \sum a_\ell X_\ell(\theta)$, where the coefficients $a_\ell$ are bounded. This expansion allows the translation of conditions on eigenvalues into conditions on the angles $\theta_k(p)$ associated with the Satake parameters.

2. Handling Exceptional Eigenvalues:
   A critical innovation is the construction of a sieve function $G(u_k)$ that accounts for primes where the GRC might fail. The authors define a set of "exceptional" primes where the angle $\theta_k(p)$ is purely imaginary (indicating $|\alpha_{u_k, j}(p)| \neq 1$).

   • They define a threshold $\delta_p$ and a parameter $L_p$ dependent on $\log T$.
   • The function $G(u_k)$ is constructed as a product of $g_p(\theta_k(p))$ minus a correction term involving $\lambda_k(qH)^2/H^2$.
   • Through a case analysis (Cases 1, 2, and 3), the authors demonstrate that if $\theta_k(p)$ falls outside the "good" range $[0, 2\pi\delta_p)$, the term $G(u_k)$ becomes non-positive. This effectively filters out forms with large exceptional eigenvalues.

3. Kuznetsov Trace Formula:
   To estimate the average value of $G(u_k)$ over the family of Maass forms $H_T = \{u_k : t_k \leq T\}$, the authors apply the truncated Kuznetsov trace formula (Lemma 2.1). This allows them to relate the sum of weighted eigenvalues to the main term (the product of coefficients $a_{0, p_j}$) and error terms involving the spectral gap.

4. Symmetric Power Coefficients:
   For the symmetric power $L$-functions, the authors express the coefficients $\lambda_{\text{sym}^m u_k}(p^a)$ as symmetric polynomials in the Satake parameters. They show that for small angles $\theta_k(p)$ (within the sieve's "good" range), these coefficients remain positive, thereby ensuring that the first sign change occurs only after the sieve's range.

Key Contributions and Results
The paper establishes an omega result for the first sign change of coefficients for Hecke-Maass cusp forms and their symmetric powers.

• Theorem 1.1: For sufficiently large $T$, there exist at least
  $$\gg |H_T| \exp\left( - \frac{(\log T) \log \log \log T}{m (\log \log T)^2} \right)$$
  Hecke-Maass forms $u_k \in H_T$ such that the least positive integer $n$ satisfying $\lambda_{\text{sym}^m u_k}(n) < 0$ satisfies:
  $$n_{\text{sym}^m u_k} \gg \frac{\log T}{m \log \log T}.$$

• Extension of Lamzouri's Work: This result generalizes Lamzouri's [6] lower bound for holomorphic forms to the Maass case. Crucially, it achieves this without assuming the Generalized Ramanujan Conjecture. The methodology successfully isolates forms with "bad" Satake parameters (those violating GRC) and shows that a significant proportion of forms still exhibit the expected behavior regarding the first sign change.

• Symmetric Powers: The paper proves that similar lower bounds hold for the coefficients of symmetric power $L$-functions, extending the scope beyond the standard Hecke eigenvalues ($m=1$).

Significance and Claims
The authors position this work as a necessary extension of recent results in the field of analytic number theory. The significance lies in:

1. Removing GRC Assumptions: By proving the result without assuming GRC, the paper provides a more robust understanding of the distribution of sign changes for Maass forms, which are less understood than their holomorphic counterparts.
2. Technical Refinement: The paper claims to refine the sieve used in [6] to specifically address the technical difficulties arising from the potential failure of GRC in the Maass setting.
3. Generalization: It demonstrates that the "omega" behavior (large lower bounds for the first sign change) is a persistent feature across symmetric power $L$-functions, not just the base $L$-function.

The paper does not propose new applications or future experimental directions but strictly focuses on establishing the theoretical lower bound for the first sign change in these specific families of $L$-functions.