Sato-tate Theorems for Mellin Transforms over Finite Fields

@inproceedings{Katz2010SatotateTF,
  title={Sato-tate Theorems for Mellin Transforms over Finite Fields},
  author={Nicholas M. Katz},
  year={2010}
}
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” (1/2π) √ 4− x2dx on the closed interval [−2, 2], and asked if this equidistribution was provably true. Rudnick had done numerics on the sums T (χ) := −(1/√q) ∑ t∈k×,t 6=1 ψ((t+ 1)/(t− 1))χ(t) 

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