A Tauberian theorem related to Weyl's criterion

AbstractLet {xn} be a sequence of real numbers and let a(n) be a sequence of positive real numbers, with A(N) = Σn=1Na(n). Tsuji has defined a notion of a(n)-uniform distribution mod 1 which is related to the problem of determining those real numbers t0 for which A(N)−1 Σn=1N a(n)e−it0xn → 0 as N → ∞. In case f(s) = Σn=1∞ a(n)e−sxn, 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)−1f(σ + it0) → 0 as σ → 0 + if and only if A(N)−1 Σn=1N a(n)e−it0xn → 0 as N → ∞