In the following, let mean all primes not exceeding n. Mertens' first theorem: does not exceed 2 in absolute value for any. Mertens' second theorem: where M is the Meissel–Mertens constant. More precisely, Mertens proves that the expression under the limit does not in absolute value exceed for any. Mertens' third theorem: where γ is the Euler–Mascheroni constant.
Changes in sign
In a paper on the growth rate of the sum-of-divisors function published in 1983, Guy Robin proved that in Mertens' 2nd theorem the difference 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 − li changes sign infinitely often. No analog of the Skewes number > li) is known in the case of Mertens' 2nd and 3rd theorems.
Regarding this asymptotic formula Mertens refers in his paper to "two curious formula of Legendre", the first one being Mertens' second theorem's prototype. He recalls that it is contained in Legendre's third edition of his "Théorie des nombres", and also that a more elaborate version was proved by Chebyshev in 1851. Note that, already in 1737, Euler knew the asymptotic behaviour of this sum. Merten's diplomatically describes his proof as more precise and rigorous. In reality none of the previous proofs are acceptable by modern standards: Euler's computations involve the infinity ; Legendre's argument is heuristic; and Chebyshev's proof, although perfectly sound, makes use of the Legendre-Gauss conjecture, which was not proved until 1896 and became better known as the prime number theorem. Mertens' proof does not appeal to any unproved hypothesis, and only to elementary real analysis. It comes 22 years before the first proof of the prime number theorem which, by contrast, relies on a careful analysis of the behavior of the Riemann zeta function as a function of a complex variable. Mertens' proof is in that respect remarkable. Indeed, with modern notation it yields whereas the prime number theorem, can be shown to be equivalent to In 1909 Edmund Landau, by using the best version of the prime number theorem then at his disposition, proved that holds; in particular the error term is smaller than for any fixed integer k. A simple summation by parts exploiting the strongest form known of the prime number theorem improves this to for some.
