A Tauberian theorem related to Weyl's criterion

Abstract

Let { x n } be a sequence of real numbers and let a( n) be a sequence of positive real numbers, with A( N) = Σ n=1 N a( n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t 0 for which A( N) −1 Σ n=1 N a( n) e − it 0 x n → 0 as N → ∞. In case f( s) = Σ n=1 ∞ a( n) e − sx n , s = σ + it, is analytic in the right half-plane 0 < σ, and satisfies a certain smoothness condition as σ → 0 +, we show that f( σ) −1 f( σ + it 0) → 0 as σ → 0 + if and only if A( N) −1 Σ n=1 N a( n) e − it 0 x n → 0 as N → ∞.