Niven constant

The Niven constant , named after the Canadian-American mathematician Ivan M. Niven , is a mathematical constant from number theory . It is defined as the limit of the arithmetic mean of the maximum exponents of the prime factorization of the first natural numbers for . ${\displaystyle n}$ ${\displaystyle n\to \infty }$

definition

Let be an integer with the prime factorization with and for , besides and the maximum of the exponents in the prime factorization of (sequence A051903 in OEIS ), for example the numbers with exactly the square-free numbers. The Niven constant is thus defined as ${\displaystyle m>1}$ ${\displaystyle m=p_{1}^{a_{1}}p_{2}^{a_{2}}p_{3}^{a_{3}}\cdots p_{k}^{a_{k}}}$ ${\displaystyle a_{i}>0}$ ${\displaystyle p_{i}\neq p_{j}}$ ${\displaystyle i\neq j}$ ${\displaystyle H\left(1\right)=1}$ ${\displaystyle H(m)=\max\{a_{1},...,a_{k}\}}$ ${\displaystyle m}$ ${\displaystyle m}$ ${\displaystyle H(m)=1}$

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}H(j).}$

properties

The Niven constant can be expressed using the Riemann zeta function and can be approximately calculated in this way (Niven 1969): ${\displaystyle \zeta \left(k\right)}$

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}H(j)=1+\sum _{k=2}^{\infty }{\biggl (}1-{\frac {1}{\zeta (k)}}{\biggr )}}$ ${\displaystyle =1{,}70521{\text{ }}11401{\text{ }}05367{\text{ }}76428{\text{ }}85514{\text{ }}53434{\text{ }}50816{\text{ }}07620{\text{ }}27651{\text{ }}65346{\text{ }}...}$(Follow A033150 in OEIS )

At the suggestion of Erdős , Niven proved for the asymptotic behavior of the minima of the exponents

${\displaystyle \sum _{j=1}^{n}h(j)=n+{\frac {\zeta ({\tfrac {3}{2}})}{\zeta (3)}}{\sqrt {n}}+o({\sqrt {n}}),}$

where and the minimum of the exponents in the prime factorization of (sequence A051904 in OEIS ) and is a Landau symbol . Thus, in particular ${\displaystyle h\left(1\right)=1}$ ${\displaystyle h(m)=\min\{a_{1},...,a_{k}\}}$ ${\displaystyle m}$ ${\displaystyle o}$

${\displaystyle \lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}h(j)=1.}$

literature

• Steven R. Finch: Niven's constant . Chapter 2.6 in Mathematical constants . Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , pp. 112-115 (English)

Web links

• Eric W. Weisstein : Niven's Constant . In: MathWorld (English).
• Sequence A033151 in OEIS ( continued fraction expansion of Niven constant)

Individual evidence

1. ↑ Ivan Niven : Averages of exponents in factoring integers . (June 18, 1968), Proceedings of the AMS 22, 1969, pp. 356-360 (English)