William,

I wonder whether I could appeal to you to make a computation– but only if it is really dead-easy and doesn’t waste too much of your time. I’m supposed to give a lecture in Princeton shortly (in the Katz conference) and it is going to be a general overview of Katz’s work. Now I would love to have some numerical displays which ”illustrate” some of the problems that Katz worked on and contributed to. For example, the distribution of angles of Kloosterman sums.

Specifically, let a be an integer, and p a prime number. Define

Kl(p, a) := ∑_{xy=a; x,y modp} exp (\frac{2πi}{p}(x + y)) .

These are real numbers, and using the Weil bounds if we write

Kl(p, a) = 2\sqrt{p} cos θ(p, a)

we might try to study the variation of the angles θ(p, a), either as p varies or as a varies, or both. One expects these angles, viewed as points in [0, π], for fixed nonzero a, varying over all primes p, to be equidistributed with respect to the Sato-Tate measure \frac{2}{π} sin^{2}θdθ. (This isn’t yet proved, if I understand right, but things ”close to it”– e.g., various averagings over choices of a– are what Katz proves

Would it be possible, for specific choices of a, to make some sort of graph of these angles for p ≤ 100 or so, demonstrating approximations to Sato Tate distributions? Needless to say, I’d be eternally grateful!