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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
$$
{\rm Kl}(p,a):=\sum_{xy=a;\;x,y \ {\rm mod} p }\exp\left({2\pi
i\over p}(x+y)\right).$$
These are real numbers, and using the Weil bounds if we write
$$
{\rm Kl}(p,a)= 2{\sqrt p}\ {\rm cos}\ \theta(p,a)$$
we might try to study the variation of the angles $\theta(p,a)$, either as
$p$ varies or as $a$
varies, or both. One expects these angles, viewed as points in $[0,\pi]$,
for fixed nonzero $a$, varying over all primes $p$, to be
equidistributed with respect to the Sato-Tate measure ${2\over \pi}{\rm
sin}^2\theta d\theta$. (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 $p \le 100$ or so, demonstrating approximations
to Sato\_Tate distributions? Needless to say, I'd be eternally grateful!
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