| $\tfrac{7}{64}=0.109375$ | [KS2003] (over $\mathbb Q$), extended uniformly to all number fields by [BB2011] | **Best known general bound.** In Sarnak’s notation, this controls both finite-place Satake parameters and archimedean spectral parameters for $\mathrm{GL}\_2(\mathbb A\_{\mathbb Q})$ (and Blomer–Brumley extend it to arbitrary number fields). |text edited 2026-08-24 — bound unchanged
Lower bound, last-listed row
| $0$ | Trivial | Conjectured to be sharp (Generalized Ramanujan / Selberg). |tracked since 2026-07-24 — no movement seen yet
Cite this rowBounds Ledger — $\mathrm{GL}_2$ Ramanujan conjecture exponent (56a)
upper bound: $\tfrac{7}{64}=0.109375$ (text edited 2026-08-24 — bound unchanged)
lower bound: $0$ (tracked since 2026-07-24 — no movement seen yet)
Mirrored from teorth/optimizationproblems@2c1968c. These are the last-listed rows of that constant's bounds table — a listing position, not a statement that this bound is the strongest or most recent. A snapshot at that sha, not a live read.
https://u00dxk2.github.io/bounds-ledger/c/56a.html