Expansion of \(\sum _{p\leq x}\log\) kp/p (Q1098876)

Using the prime number theorem and partial summation the author shows that \[ \sum_{p\leq x}\log \quad kp\cdot p^{-1}=k^{-1} \log \quad kx+E_ k+O((\log x)^{k+1} \quad \exp (-c(\log x)^{1/10})) \] and gives a representation of the constants \(E_ k\) which employs the expansion of \(\log (\zeta (s)\cdot (s-1))\) into powers of \(s-1.\) Reviewer's remark: The asymptotic formula mentioned above is a special case of Landau's general result on \(\sum_{p\leq x}F(p)\) (cf. {\S} 55 of Landau's \textit{Handbuch}).