On Ramanujan Primes for Hecke-Maass Cusp Forms

Technical Summary: On Ramanujan Primes for Hecke-Maass Cusp Forms

Problem Statement
The Ramanujan conjecture for a primitive Hecke-Maass cusp form $\phi$ asserts that for any prime $p$ not dividing the level $N$, the $p$-th Hecke eigenvalue $\lambda_\phi(p)$ satisfies the sharp bound $|\lambda_\phi(p)| \le 2$. While this remains an open problem in general (with the best known bound being $|\lambda_\phi(p)| \le p^{7/64} + p^{-7/64}$ due to Kim and Sarnak), Luo and Zhou [LZ19] previously established that the set of primes $RP(\phi) = \{p \nmid N : |\lambda_\phi(p)| \le 2\}$ has a lower natural density of at least $34/35$. Furthermore, they showed the existence of a least prime in $RP(\phi)$ bounded by a power of the analytic conductor.

This paper extends these investigations to the context of multiple distinct primitive Hecke-Maass cusp forms. The authors address two primary questions:

1. Simultaneous Occurrence: What is the upper bound for the least prime $p$ at which the Ramanujan conjecture holds simultaneously for two or three distinct forms?
2. Union Density: What is the lower natural density of the set of primes at which the Ramanujan conjecture holds for at least one form in a given finite (or infinite) family of distinct forms?

Methodology
The paper employs analytic number theory techniques centered on the properties of $L$-functions associated with the cusp forms.

• For Least Prime Bounds (Theorems 1 & 2):
  The authors utilize the method of "sieving" combined with the analytic properties of adjoint square $L$-functions, $L(s, \text{Ad}(\phi))$.

  • They define a summation $S(x)$ involving the Dirichlet coefficients of the product of adjoint $L$-functions.
  • A crucial observation is that if the Ramanujan conjecture fails at $p$ (i.e., $|\lambda_\phi(p)| > 2$), then the twisted Hecke eigenvalue $\lambda_\phi(p^2)\chi_\phi(p)$ must be large (specifically $> 3$).
  • By constructing a specific multiplicative function $h(n)$ and utilizing the convexity bounds for $L(s, \text{Ad}(\phi))$, the authors derive an upper bound for $S(x)$ via contour integration and a lower bound via the positivity of the sieve weights.
  • The proof relies on the non-vanishing of specific linear combinations of eigenvalues. For two forms, the sum of their twisted eigenvalues is shown to be positive if the conjecture fails for both; for three forms, a similar but distinct linear combination is used. The authors note that this specific argument fails for four or more forms because the positivity of the sum cannot be guaranteed.

• For Density Bounds (Theorems 3 & 4):
  The approach shifts to analyzing the distribution of primes using Rankin-Selberg $L$-functions and zero-free regions.

  • The authors introduce the twisted symmetric fourth power $A_4(\phi) = \text{Sym}^4(\phi) \otimes \chi_\phi^2$ and the symmetric third power $\text{Sym}^3(\phi)$.
  • They establish that for distinct forms $\phi_1, \phi_2$, the associated automorphic representations (specifically $\text{Ad}(\phi)$ and $A_4(\phi)$) are distinct.
  • Leveraging recent results by Thorner and Zhao [TZ26] regarding the absence of Landau-Siegel zeros for Rankin-Selberg $L$-functions of the form $L(s, \text{Ad}(\phi_1) \times \text{Ad}(\phi_2))$ and $L(s, \text{Ad}(\phi_1) \times A_4(\phi_2))$, the authors prove that the cross-correlation sums of the coefficients $A_\phi(p)$ and $A^{[4]}_\phi(p)$ are $o(\pi(X))$.
  • They construct a non-negative polynomial $U(p)$ in terms of these coefficients. By expanding $U(p)$ and applying the Prime Number Theorem for automorphic $L$-functions, they bound the density of primes where the Ramanujan conjecture fails for all forms in the set.

Key Contributions and Results

1. Upper Bounds for Least Simultaneous Ramanujan Primes:

   • Theorem 1: For two distinct primitive Hecke-Maass cusp forms $\phi_1, \phi_2$, there exists a prime $p \in RP(\phi_1) \cap RP(\phi_2)$ such that:
     $$p \ll [N_1 N_2 (1+|t_{\phi_1}|)(1+|t_{\phi_2}|)]^{0.447374}$$
   • Theorem 2: For three distinct forms $\phi_1, \phi_2, \phi_3$, there exists a prime $p \in \bigcap_{i=1}^3 RP(\phi_i)$ such that:
     $$p \ll [N_1 N_2 N_3 \prod_{i=1}^3 (1+|t_{\phi_i}|)]^{0.778798}$$
   • The authors explicitly state that their method does not extend to four or more forms due to the inability to guarantee the positivity of the relevant eigenvalue sums.

2. Improved Lower Bounds for Natural Density:

   • Theorem 3: For two distinct forms $\phi_1, \phi_2$ (assuming their sets of Ramanujan primes are not identical up to a finite set), the lower natural density of the union satisfies:
     $$\underline{d}(RP(\phi_1) \cup RP(\phi_2)) \ge \frac{43}{44}$$
     This improves upon the trivial bound of $34/35$ derived from individual densities.
   • Theorem 4: For a finite family of $m$ distinct forms $\{\phi_i\}_{i=1}^m$, the lower natural density of the union is bounded by:
     $$\underline{d}\left(\bigcup_{j=1}^m RP(\phi_j)\right) \ge 1 - \frac{1}{26 + 9m}$$
   • Corollary 1: For an infinite family of distinct forms, the natural density of the union is exactly 1. This implies that for almost all primes (in the sense of natural density), the Ramanujan conjecture holds for at least one form in the family.

Significance and Claims
The paper claims to advance the understanding of the distribution of primes satisfying the Ramanujan conjecture in a multi-form setting. By moving from single-form analysis to simultaneous and union analyses, the authors demonstrate that the "bad" primes (where the conjecture fails) become increasingly sparse when considering multiple forms.

The significance lies in:

• Providing explicit quantitative bounds for the first occurrence of simultaneous Ramanujan primes for pairs and triples of forms.
• Establishing that the union of Ramanujan prime sets for distinct forms covers a significantly larger proportion of the primes than individual sets, approaching density 1 as the family size increases.
• Utilizing the non-vanishing of specific Rankin-Selberg $L$-functions (via [TZ26]) to refine density estimates, a technique the authors generalize from two forms to arbitrary finite families.

The authors remain modest regarding the limitations of their current method, noting that the extension to four or more forms for the least prime bound is currently obstructed by the failure of the positivity argument for the relevant eigenvalue sums. Similarly, they note that further improving the density bound for finite families (specifically the coefficient $26+9m$) would require establishing zero-free regions for $L(s, A_4(\phi_1) \times A_4(\phi_2))$, which is currently beyond reach.