# Mertens’ first theorem

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

 $\sum_{p\leq x}\frac{\ln p}{p}=\ln x+O(1)$

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 MertensFirstTheorem 2013-03-22 11:46:06 2013-03-22 11:46:06 KimJ (5) KimJ (5) 8 KimJ (5) Theorem msc 11A25 msc 17B45 msc 17B66