Generalised Ramanujan–Petersson conjecture for Hecke–Maass cusp forms

Prove the Generalised Ramanujan–Petersson Conjecture for Hecke–Maass cusp forms on GL_n by establishing the pointwise bound A(m,1,\ldots,1)\ll_{\varepsilon} m^{\varepsilon} for every \varepsilon>0 and every n\geq 2.

Background

The Generalised Ramanujan–Petersson Conjecture is used throughout the paper as an assumption for the main results on signs of Hecke eigenvalues in short intervals and for higher-rank forms. It predicts the optimal pointwise growth bound for the Hecke eigenvalues and, equivalently, for the Satake parameters.

The paper notes that only partial bounds are currently known, with an exponent \vartheta(n)>0 in general, while the conjecture predicts that \vartheta(n)=0 is admissible for every rank n\geq 2. Thus the conjecture remains unresolved in the general higher-rank setting.

References

In contrast, the Ramanujan conjecture predicts that the eigenvalue $\lambda_{\varpi,a}(p)$ of $T_{a}(p)$ on each irreducible cuspidal subspace $V_\varpi\subseteq L2(\Gamma G)$ is $O_a(p{A/2})$.

— The non-spherical sup-norm problem for $\mathrm{GL}(n)$ and generalized spherical functions  (2609.19121 - Blomer et al., 16 Sep 2026) in Section 6.1, “Hecke operators”

Although the Ramanujan conjecture predicts that cuspidal representations of $\GL(n)$ are tempered, there exist counterexamples for $\Sp(4)$ .

— Automorphic $C^*$-algebras of reductive groups  (2609.35543 - Yang, 28 Sep 2026) in Section 1, Introduction

Generalised Ramanujan--Petersson Conjecture predicts that an estimate of the form $A(m,1,...,1)\ll_ m$ holds for every $>0$.

— On the Number of Hecke Eigenvalues of Same Sign on $\mathrm{GL}_n$  (2609.10446 - Jääsaari, 9 Sep 2026) in Section 3, subsection “Automorphic forms,” paragraph beginning “This result can be interpreted...”