In the following, let mean all primes not exceeding n.
Mertens' first theorem:
does not exceed 2 in absolute value for any . (A083343)
Mertens' second theorem:
for any .
Mertens' third theorem:
Changes in sign
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.
Mertens' second theorem and the prime number theorem
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. In reality none of the previous proofs are acceptable by modern standards: Euler's computations involve the infinity (and the hyperbolic logarithm of infinity, and the logarithm of the logarithm of 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 (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. Mertens' proof is in that respect remarkable. Indeed, with modern notation it yields
whereas the prime number theorem (in its simplest form, without error estimate), can be shown to be equivalent to
for some .
Similarly a partial summation shows that is equivalent to the PNT.
Mertens' third theorem and sieve theory
An estimate of the probability of () having no factor is given by
This is closely related to Mertens' third theorem which gives an asymptotic approximation of
The main step in the proof of Mertens' second theorem is
where the last equality needs which follows from .
Thus, we have proved that
A partial summation yields
- F. Mertens. J. reine angew. Math. 78 (1874), 46–62 Ein Beitrag zur analytischen Zahlentheorie
- 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.
- P.L. Tchebychev. Sur la fonction qui détermine la totalité des nombres premiers. Mémoires présentés à l'Académie Impériale des Sciences de St-Pétersbourg par divers savants, VI 1851, 141–157
- Although this equivalence is not explicitly mentioned there, it can for instance be easily derived from the material in chapter I.3 of: G. Tenenbaum. Introduction to analytic and probabilistic number theory. Translated from the second French edition (1995) by C. B. Thomas. Cambridge Studies in Advanced Mathematics, 46. Cambridge University Press, Cambridge,1995.
- Edmund Landau. Handbuch der Lehre von der Verteilung der Primzahlen, Teubner, Leipzig 1909, Repr. Chelsea New York 1953, § 55, p. 197-203.
- Yaglom and Yaglom Challenging mathematical problems with elementary solutions Vol 2, problems 171, 173, 174