Higher-rank analogue of Serre’s estimate for vanishing Hecke eigenvalues

Establish an analogue for higher-rank Hecke–Maass cusp forms of Serre’s estimate controlling the size of the set of primes p for which the Hecke eigenvalue A(p) vanishes.

Background

The paper studies the signs of the real Hecke eigenvalues A(m) of self-dual Hecke–Maass cusp forms on GL_n. Existing methods for proving many positive and negative eigenvalues in the holomorphic GL_2 setting use the theory of B-free numbers and require an estimate for the number of primes p with A(p)=0.

For higher-rank Hecke–Maass cusp forms, the paper states that an analogue of Serre’s estimate is not known. Such an estimate is described as a challenging open problem because it would provide the non-lacunarity input needed to extend the B-free-number argument and, in particular, would yield the expected lower bound for the number of positive and negative Hecke eigenvalues.

References

The main reason for this weaker result in the setting of (higher rank) Maass cusp forms is that the method of $\mathcal B$-free numbers used in the case of holomorphic cusp forms is difficult to generalise as it uses Serre's estimate on the size of the set ${p\leq x\,:\,A(p)=0}$ as an input, a analogue of this result for (higher rank) Maass cusp forms is not known and is a challenging open problem.

— On the Number of Hecke Eigenvalues of Same Sign on $\mathrm{GL}_n$  (2609.10446 - Jääsaari, 9 Sep 2026) in Introduction, paragraph following the discussion of the lower bound of Liu and Wu