In the following, let mean all primes not exceeding n.
Mertens' 1st theorem:
does not exceed 2 in absolute value for every .
Mertens' 2nd theorem:
where M is the Meissel–Mertens constant. More precisely, Mertens proves (loc. cit.) that the expression under the limit does not in absolute value exceed
for every .
Mertens' 3rd theorem:
where γ is the Euler–Mascheroni constant.
changes sign infinitely often, and that in Mertens' 3rd theorem the difference
changes sign infinitely often. Robin's results are analogous to Littlewood's famous theorem that the difference π(x) − li(x) changes sign infinitely often. No analog of the Skewes number (an upper bound on the first natural number x for which π(x) > li(x)) is known in the case of Mertens' 2nd and 3rd theorems.
converges to A and another
- Robin, G. (1983). "Sur l’ordre maximum de la fonction somme des diviseurs". Séminaire Delange–Pisot–Poitou, Théorie des nombres (1981–1982). Progress in Mathematics 38: 233–244.
Further reading 
- Yaglom and Yaglom Challenging mathematical problems with elementary solutions Vol 2, problems 171, 173, 174