Hardy and Ramanujan theorem

The set of Hardy and Ramanujan from number theory states that the number of distinct prime factors of a whole number , the normal order has. The theorem was proven in 1917 by Godfrey Harold Hardy and S. Ramanujan . ${\displaystyle \omega (n)}$ ${\displaystyle n}$ ${\displaystyle \log(\log(n))}$

An arithmetic function is of normal magnitude if holds for each ${\displaystyle f(n)}$ ${\displaystyle g(n)}$ ${\displaystyle \varepsilon >0}$

${\displaystyle (1-\varepsilon )g(n)\leq f(n)\leq (1+\varepsilon )g(n)}$

for almost all n, that is, the proportion of those for which the inequality does not apply, tends towards zero. ${\displaystyle n<x}$ ${\displaystyle x\to \infty }$

The following applies more precisely:

The number of for which ${\displaystyle n\leq x}$

${\displaystyle |\omega (n)-\log(\log(n))|>{(\log(\log(n)))}^{{\frac {1}{2}}+\delta }}$

holds, is of the order for each (with the Landau symbols , that is, the proportion of those for which the inequality holds, vanishes asymptotically for ). ${\displaystyle \delta >0}$ ${\displaystyle o(x)}$ ${\displaystyle n\leq x}$ ${\displaystyle x\to \infty }$

The proof can be found in the Hardy and Wright number theory textbook. Pál Turán gave a simplified proof in 1934. Turán extended the theorem to include other strongly additive, arithmetic functions.

Hubert Delange proved in 1953 that essentially has a Gaussian normal distribution, which is also the subject of Erdős-Kac's theorem . ${\displaystyle \omega (n)}$

The theorem also applies to the function in which the prime factors are summed with their multiplicity. ${\displaystyle \Omega (n)}$

Individual evidence

1. ↑ Hardy, Ramanujan: The normal number of prime factors of a number n. In: Quarterly Journal of Mathematics. Volume 48, 1917, pp. 76-92.
2. ↑ ^{a b} Hardy, Wright: An Introduction to number theory. Oxford University Press, 1975, p. 356. Translation: Introduction to Number Theory . Oldenbourg, Munich 1958, p. 404.
3. ↑ Hardy, Wright: Introduction to Number Theory . Oldenbourg, Munich 1958, p. 405.
4. ^ Pál Turán: On a theorem of Hardy and Ramanujan. In: Journal of the London Mathematical Society. Volume 9, 1934, pp. 274-276.
5. ↑ Delange, Compte Rend. Acad. Sci. Paris, Volume 237, 1953, pp. 543-544. According to Halberstam: About additive number theoretic functions. J. Reine Angewandte Mathematik, Volume 195, 1955, p. 210, SUB Göttingen .
6. ↑ Erdös, Kac, Am. J. Math., Volume 38, 1940, pp. 738-742, and Alfred Renyi, Pal Turan: On a theorem of Erdös-Kac. Acta Arithmetica, Volume 4, 1958, pp. 71-84, digitized .