For every integer n≥ 2 and every cuspidal automorphic representation pi of GL_n over a number field, every unramified local component pi_v is tempered. Equivalently, if alpha_pi,1(v),…,alpha_pi,n(v) are the Satake parameters at an unramified place v, then |alpha_pi,j(v)|=1 for every j. In particular, for a self-dual Hecke--Maass cusp form on SL_n(Z), all unramified Satake parameters have absolute value 1.

Generalised Ramanujan–Petersson conjecture

The Generalised Ramanujan–Petersson conjecture is a broad coefficient-control assertion for automorphic forms. The sources record conditional consequences and a recent sign-distribution advance, but no verified proof or counterexample.

Known results

Jiseong Kim obtains conditional short-sum, shifted-convolution, and sign-change estimates for Hecke–Maass coefficients, assuming the conjecture and suitable zero-free regions.
An arXiv paper claims the rank-two Ramanujan–Petersson conjecture for Maass forms on $\mathrm{SL}(2,\mathbb{Z})$ , but this does not address the full $\mathrm{GL}_n$ conjecture.

September 2026 sign-distribution bounds

On September 9, 2026, On the Number of Hecke Eigenvalues of Same Sign on $\mathrm{GL}_n$ reported stronger short- and long-interval sign-distribution bounds, including unconditional low-rank results. Its main short-interval result remains conditional, and the paper is unrefereed; it does not claim to prove the conjecture.

Current status (as of September 2026): The general conjecture remains open; recent work gives conditional consequences and some unconditional low-rank sign-distribution results, not a proof or disproof.

Generalised Ramanujan–Petersson conjecture

Primary source

Additional references

Progress summary

Solutions 0