Let k be a finite field, q its cardinality, p its characteristic,
Ã : (k, +) → Z[ζp]× ⊂ C×
a nontrivial additive character of k, and
χ : (k×, ×) → Z[ζq–1]× ⊂ C×
a (possibly trivial) multiplicative character of k.
The present work grew out of two questions, raised by Ron Evans and Zeev Rudnick respectively, in May and June of 2003. Evans had done numerical experiments on the sums
as χ varies over all multiplicative characters of k. For each χ, S(χ) is real, and (by Weil) has absolute value at most 2. Evans found empirically that, for large q = #k, these q – 1 sums were approximately equidistributed for the “Sato-Tate measure” ...