Niven's constant

In number theory, Niven's constant, named after Ivan Niven, is the largest exponent appearing in the prime factorization of a natural number "on average". More precisely, if we set $H(1)=1$ and define $H(n)$ to be the largest exponent appearing in the prime factorization of $n>1$, then Niven's constant is given by

$\lim _{n\to \infty }{\frac {1}{n}}\sum _{j=1}^{n}H(j)=1+\sum _{k=2}^{\infty }{\bigg (}1-{\frac {1}{\zeta (k)}}{\bigg )}=1.7052111401\dots$

where $\zeta$ is the Riemann zeta function. Niven also proved that

$\sum _{j=1}^{n}h(j)=n+c{\sqrt {n}}+o({\sqrt {n}}),$

where $h(1)=1$, $h(n)$ is defined to be the smallest exponent appearing in the prime factorization of $n>1$, $o$ is little-o notation, and $c=\zeta (\textstyle {\frac {3}{2}})/\zeta (3)$. Consequently,

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

References

• Finch, Steven R. (2003). Mathematical Constants. Encyclopedia of Mathematics and Its Applications. Vol. 94. Cambridge University Press. ISBN 0-521-81805-2.

• Niven, Ivan (1969). "Averages of exponents in factoring integers". Proceedings of the American Mathematical Society. 22 (2): 356–360. JSTOR 2037055.

External links

• Sloane, N. J. A. (ed.). "Sequence A033150". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

• Weisstein, Eric W. "Niven's constant". MathWorld.