Mertens theorem (number theory)

The Mertens theorem , named after the mathematician Franz Mertens , is understood in the mathematical branch of analytical number theory to mean a series of statements about the asymptotic behavior of series that are formed from reciprocal values ​​of the prime numbers .

Definitions

To formulate the statements, we first recall the O notation with which the growth of functions (with the help of a function and where there is a constant) can be compared. This is how you write , if there is a constant and a , so that for all . The functions appearing in the formulas of Mertens are the natural logarithm , with designated Mangoldt function and Chebyshev function ${\displaystyle f,g:[K;+\infty )\rightarrow \mathbb {R} }$ ${\displaystyle h:[K;+\infty )\rightarrow \mathbb {R} _{*}^{+}}$ ${\displaystyle K}$ ${\displaystyle f(x)=g(x)+O(h(x))}$ ${\displaystyle C>0}$ ${\displaystyle x_{0}>0}$ ${\displaystyle |f(x)-g(x)|\leq C\cdot h(x)}$ ${\displaystyle x>x_{0}}$ ${\displaystyle \log }$ ${\displaystyle \Lambda }$

${\displaystyle \psi (x)=\sum _{n\leq x}\Lambda (n)=\sum _{p^{n}\leq x}\log p}$.

It runs through all prime numbers and all natural numbers; Totaling is only made for those or and for which the condition specified under the total symbol is met. This abbreviated form is also used in the formulas presented here. ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle p}$

Mertens' formulas

The following relationships apply to the functions mentioned above.

${\displaystyle (1)\quad \sum _{n\leq x}{\frac {\Lambda (n)}{n}}=\log x\,+\,O(1).}$

${\displaystyle (2)\quad \sum _{p\leq x}{\frac {\log p}{p}}=\log x\,+\,O(1).}$

${\displaystyle (3)\quad \int _{1}^{x}{\frac {\psi (t)}{t^{2}}}\mathrm {d} t=\log x\,+\,O(1).}$

There's a constant , so that ${\displaystyle M}$

${\displaystyle (4)\quad \sum _{p\leq x}{\frac {1}{p}}=\log \log x+M\,+\,O\left({\frac {1}{\log x}}\right).}$

${\displaystyle M}$is the Meissel-Mertens constant and the following applies:

${\displaystyle M=\lim _{n\to \infty }\left(\sum _{p\in \mathbb {P} }^{n}{\frac {1}{p}}-\log \log n\right)=\gamma +\sum _{p\in \mathbb {P} }\left[\log \left(1-{\frac {1}{p}}\right)+{\frac {1}{p}}\right]=\gamma +\sum _{k=2}^{\infty }{\frac {\mu (k)}{k}}\log \zeta (k).}$

The Euler-Mascheroni constant denotes and it applies ${\displaystyle \gamma =0,5772156649...}$

${\displaystyle (5)\quad \lim _{x\to \infty }\log x\prod _{p\leq x}\left(1-{\frac {1}{p}}\right)=e^{-\gamma }.}$

Remarks

The original formulas from Mertens are (2), (4) and (5). Mertens calls less precise versions of (4) and (5) "Strange Formulas" by Legendre . Euler knew that the series of reciprocal values ​​of all prime numbers diverges (like ) . The formula (4) describes precisely how quickly this series diverges towards infinity. The last formula is a consequence of this, as shown in the cited Hardy and Wright textbook. The formulas were first proven by Franz Mertens in 1874. Formula (4) was recognized by Chebyshev , but his proof used the Legendre-Gauss conjecture, which could not be shown until 1896 and which then became known as the prime number theorem. Mertens, however, did not use an unproven conjecture (in 1874). His evidence is remarkable for two reasons. Mertens had the idea to prove (2) first, with (4) following relatively easily. Second, we now know that formula (4) is "almost" equivalent to the prime number theorem: in fact, it is equivalent to ${\displaystyle \log \log x}$

${\displaystyle (4')\quad \sum _{p\leq x}{\frac {1}{p}}=\log \log x+C\,+\,o\left({\frac {1}{\log x}}\right).}$

Individual evidence

1. ↑ K. Chandrasekharan : Introduction to Analytic Number Theory , Springer Verlag, Grundlehren der Mathematischen Wissenschaften 148, ISBN 3540041419 , VII, §5, Theorem 8
2. ^ ^{A b} G. H. Hardy , EM Wright : An Introduction to the Theory of Numbers , 4th edition, Oxford University Press, Oxford 1975. ISBN 0-19-853310-1 , Theorem 429
3. ↑ Essai sur la théorie des nombres ; 3rd edition (1830), two volumes; German Leipzig 1886. Fourth part, VIII.
4. ^ Leonhard Euler. Variae observationes circa series infinitas. Commentarii academiae scientiarum Petropolitanae 9 (1737), 160-188.
5. ↑ F. Mertens. J. pure angew. Math. 78 (1874), 46-62. A contribution to analytical number theory
6. ^ PL 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
7. ↑ Although this equivalency is not specifically mentioned, it is easy to get hold of. For example with I.3 by: G. Tenenbaum. Introduction à la théorie analytique et probabiliste des nombres. Cours spécialisés 1st Société Mathématique de France, Paris 1995.