Kloosterman vs. Salié

I’ve just realized a rather obvious fact concerning the vague question at the end of my latest post on Kloostermania, which I’ll rephrase informally:

Can one guess with better than even chance that a graph like Kloostermania’s represents a Kloosterman or a Salié sum?

I had said that, for fixed p, just knowing the sum shouldn’t leave much room to make any choice except a random one. To make sense, the rule must be made more precise: you are given a bare real number, say

and you are told that it is either the value of a Kloosterman sum S(1,1;p) for some prime, or of a Salié sum T(1,-1;p) for some prime. You can win a drink of your choice by picking up the right one. What can you say? The reason it is hard to do better than heads-Kloosterman/tail-Salié is that you do not know p. If you knew the value of the prime, then you could compute the angle in [0,π] such that

and use the fact that these are supposed to be distributed differently for Kloosterman and Salié sums. For instance, the probability that θ is(conjecturally) larger than 3π/4 is bigger for Salié sums (it is 1/4) than for Kloosterman sums (about 0.09…), and hence, finding an angle in this range would give a big hint that it comes from a Salié sum (but no certainty, of course).

And now for the really obvious point: if you can, in addition to knowing S, actually see the graph of the partial sums, then — of course — you know the prime: you just have to count the steps on the graph.

It seems that similar ideas should lead to a slightly better than average guess even without knowing p: the size of S gives a lower bound on p, and for each possible guess of p, we have a guess of either Salié or Kloosterman. Presumably, combining these should be possible to get a small gain on pure chance…

(Note: readers are welcome to make a guess concerning the value above…)