For positive integers m and N, we estimate the rational exponential sumswith denominator m over the reductions modulo m of elements of the set F(N) = {s/ r : r, s ∈ Z, gcd(r, s) = 1, N ≥ r ≥ s ≥ 1} of Farey fractions of order N (only fractions s/r with gcd(r, m) = 1 are considered).
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