Mertens' theorems

In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens. In the following, let mean all primes not exceeding n. Mertens' first theorem is that does not exceed 2 in absolute value for any . () Mertens' second theorem is 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 . The main step in the proof of Mertens' second theorem is where the last equality needs which follows from . Thus, we have proved that Since the sum over prime powers with converges, this implies A partial summation yields 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(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. Regarding this asymptotic formula Mertens refers in his paper to "two curious formula of Legendre", the first one being Mertens' second theorem's prototype (and the second one being Mertens' third theorem's prototype: see the very first lines of the paper). He recalls that it is contained in Legendre's third edition of his "Théorie des nombres" (1830; it is in fact already mentioned in the second edition, 1808), 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. Mertens diplomatically describes his proof as more precise and rigorous.