Mertens’ first theorem

For any real number $x\ge 2$ we have

$$\sum _{p\le x}\frac{\ln p}{p}=\ln x+O(1)$$

for all prime integers $p$.

Moreover, the term $O(1)$ arising in this formula lies in the open interval $(-1-\ln 4,\ln 4)$.

Title	Mertens’ first theorem
Canonical name	MertensFirstTheorem
Date of creation	2013-03-22 11:46:06
Last modified on	2013-03-22 11:46:06
Owner	KimJ (5)
Last modified by	KimJ (5)
Numerical id	8
Author	KimJ (5)
Entry type	Theorem
Classification	msc 11A25
Classification	msc 17B45
Classification	msc 17B66