The distribution of α p modulo one 269
نویسنده
چکیده
We prove that, for any irrational number α, there are infinitely many primes p such that ‖αp‖ < p−1/3+ . Here ‖y‖ denotes the distance from y to the nearest integer. The proof uses Harman’s sieve method with arithmetical information coming from bounds for averages of Kloosterman sums.
منابع مشابه
Distribution of Farey Fractions in Residue Classes and Lang–Trotter Conjectures on Average
We prove that the set of Farey fractions of order T , that is, the set {α/β ∈ Q : gcd(α, β) = 1, 1 ≤ α, β ≤ T}, is uniformly distributed in residue classes modulo a prime p provided T ≥ p1/2+ε for any fixed ε > 0. We apply this to obtain upper bounds for the Lang–Trotter conjectures on Frobenius traces and Frobenius fields “on average” over a one-parametric family of elliptic curves.
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