 in number theory, mertens' theorems are three 1874 results related to the density of prime numbers proved by franz mertens.f. mertens. j. reine angew. math. 78 (1874), 46-62 ein beitrag zur analytischen zahlentheorie "mertens' theorem" may also refer to his theorem in analysis. == in number theory == in the following, let p\le n mean all primes not exceeding n. mertens' 1st theorem: : \sum_{p \le n} \frac{\ln p}{p} - \ln n does not exceed 2 in absolute value for any n\ge 2. mertens' 2nd theorem: :\lim_{n\to\infty}\left(\sum_{p\le n}\frac1p -\ln\ln n-m\right) =0, where m is the meisselâmertens constant. more precisely, mertens proves that the expression under the limit does not in absolute value exceed : \frac 4{\ln(n+1)} +\frac 2{n\ln n} for any n\ge 2. mertens' 3rd theorem: :\lim_{n\to\infty}\ln n\prod_{p\le n}\left(1-\frac1p\right)=e^{-\gamma}, where Î³ is the eulerâmascheroni 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 :\sum_{p\le n}\frac1p -\ln\ln n-m changes sign infinitely often, and that in mertens' 3rd theorem the difference :\ln n\prod_{p\le n}\left(1-\frac1p\right)-e^{-\gamma} changes sign infinitely often. robin's results are analogous to littlewood's famous theorem that the difference Ï(x)&nbsp;â&nbsp;li(x) changes sign infinitely often. no analog of the skewes number (an upper bound on the first natural number x for which Ï(x)&nbsp;>&nbsp;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 "thÃ©orie 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.p.l. tchebychev. sur la fonction qui dÃ©termine la totalitÃ© des nombres premiers. mÃ©moires prÃ©s
Mathematical_series Summability_theory Theorems_about_prime_numbers