Another proof on the existence of Mertens's constant (Q542530)

In this paper, the author gives a new proof of Mertens' constant; that is \[ \lim_{x\to\infty}\sum_{p\leq x}\left(\frac{1}{p}-\log\log x\right) \] exists and equals \(0.26149\cdots\). Results are obtained via estimates on the \(k\)th prime by \textit{J.~B.~Rosser} [Proc. Lond. Math. Soc. (2) 45, 21--44 (1939; Zbl 0019.39401)] and \textit{J.~B.~Rosser} and \textit{L.~Schoenfeld} [Ill. J. Math. 6, 64--94 (1962; Zbl 0122.05001)].