In analytic number theory, Mertens' theorems are three 1874 results related to the density of prime numbers proved by Franz Mertens.
More precisely, Mertens[1] 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
converges, this implies A partial summation yields In a paper [2] 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.
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.
[3] Note that, already in 1737, Euler knew the asymptotic behaviour of this sum.
Mertens diplomatically describes his proof as more precise and rigorous.
); 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 (in 1874), 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.
Indeed, with modern notation it yields whereas the prime number theorem (in its simplest form, without error estimate), can be shown to imply [4] In 1909 Edmund Landau, by using the best version of the prime number theorem then at his disposition, proved[5] 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
is given by This is closely related to Mertens' third theorem which gives an asymptotic approximation of