Statistical Properties of Eigenvalues of the Hecke Operators

Abstract

Two basic questions concerning the Ramanujan τ-function concern the size and variation of these numbers:

• (i) Ramanujan conjecture: $\left| {\tau (p)} \right| \leqslant 2\text{p}^{11/2}$ for all primes p.

• (ii) “Sato-Tate” conjecture: $\text{a}_\text{p} = \frac{{\tau (\text{p})}}{{\text{p}^{11/2} }}$ is equidistributed with respect to

  $$\text{d}\mu (\text{x}) = \left\{ \begin{gathered}\frac{1}{{2\pi }}\sqrt {4 - \text{x}^2 } \text{dx}\,\,\,\,\,\,\,\text{if}\,\,\left| \text{x} \right| \leqslant 2 \hfill \\0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{otherwise} \hfill \\\end{gathered} \right.$$

  as p → ∞. We refer to the last as the semicircle distribution.