Half-Integer Weight Ramanujan Bound Still Unproven — E8 Intelligence Research

FINDING: The Ramanujan-Petersson conjecture for half-integer weight cusp forms remains a deep open problem; the strongest known bounds for Fourier coefficients are subconvex but not optimal, while the integer-weight case (Deligne) gives the sharp \(O(n^{(k-1)/2+\epsilon})\) bound. | MATH: For a half-integer weight \(k+1/2\) cusp form \(f(z)=\sum_{n\ge1} a(n) e^{2\pi i n z}\), the conjecture states \(|a(n)| \ll n^{k/2-1/4+\epsilon}\). Current best (Iwaniec, Duke, et al.) give \(|a(n)| \ll n^{k/2-1/4-1/28+\epsilon}\) (or similar subconvex exponents). For integer weight \(k\), Deligne proved \(|a(n)| \ll n^{(k-1)/2+\epsilon}\) via Weil conjectures. Ramanujan tau function: \(\tau(n)\) satisfies \(\tau(p) \equiv 1+p^{11} \pmod{691}\), and Sato-Tate: \(\frac{\tau(p)}{2p^{11/2}} \in [-1,1]\) equidistributed w.r.t. \(\frac{2}{\pi}\sqrt{1-x^2}\,dx\). | CONNECTION: The optimal bound corresponds to the "trivial" bound from the Petersson inner product — the norm of the form gives \(|a(n)| \ll n^{k Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com