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Author: William A. Stein
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William,
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I wonder whether I could appeal to you to make a computation-- but
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only if it is really dead-easy and doesn't waste too much of your
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time. I'm supposed to give a lecture in Princeton shortly (in the Katz
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conference) and it is going to be a general overview of Katz's
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work. Now I would love to have some numerical displays which
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"illustrate" some of the problems that Katz worked on and contributed
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to. For example, the distribution of angles of Kloosterman sums.
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Specifically, let $a$ be an integer, and $p$ a prime number. Define
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$$
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{\rm Kl}(p,a):=\sum_{xy=a;\;x,y \ {\rm mod} p }\exp\left({2\pi
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i\over p}(x+y)\right).$$
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These are real numbers, and using the Weil bounds if we write
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$$
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{\rm Kl}(p,a)= 2{\sqrt p}\ {\rm cos}\ \theta(p,a)$$
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we might try to study the variation of the angles $\theta(p,a)$, either as
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$p$ varies or as $a$
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varies, or both. One expects these angles, viewed as points in $[0,\pi]$,
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for fixed nonzero $a$, varying over all primes $p$, to be
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equidistributed with respect to the Sato-Tate measure ${2\over \pi}{\rm
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sin}^2\theta d\theta$. (This isn't yet proved, if I understand right, but
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things "close to it"-- e.g., various averagings over choices of $a$-- are
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what Katz proves
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Would it be possible, for specific choices of $a$, to make some sort of
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graph of these angles for $p \le 100$ or so, demonstrating approximations
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to Sato\_Tate distributions? Needless to say, I'd be eternally grateful!
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