Note on the distribution of Ramanujan’s tau function

Abstract:

According to a conjecture of Sato and Tate, the angle $\theta$ whose cosine is $\tfrac {1} {2}\tau (p){p^{ - 11/2}}$, where $\tau$ is Ramanujan’s function and $p$ a prime, is distributed over $[0,\pi ]$ according to a ${\sin ^2}\theta$ law. The paper reports on a test of this conjecture for the 1229 primes under 10000. Extreme values of $\theta$ are also given.