Some Ω-theorems for the Riemann zeta-function

Kai-Man Tsang

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In 1989, E. Saias established an asymptotic formula for $Ψ (x, y) = |\{n \leq x : p ∣ n \Rightarrow p \leq y\}|$ with a very good error term, valid for $exp ({(log log x)}^{(5 / 3) + ϵ}) \leq y \leq x$ , $x \geq x_{0} (ϵ)$ , $ϵ > 0 .$ We extend this result to an algebraic number field $K$ by obtaining an asymptotic formula for the analogous function $Ψ_{K} (x, y)$ with the same error term and valid in the same region. Our main objective is to compare the formulae for $Ψ (x, y)$ and $Ψ_{K} (x, y),$ and in particular to compare the second term in the two expansions.