In number theory, Niven's constant, named after Ivan Niven, is the largest exponent appearing in the prime factorization of any natural number n "on average". More precisely, if we define H(1) = 1 and H(n) = the largest exponent appearing in the unique prime factorization of a natural number n > 1, then Niven's constant is given by
\limn
| 1 | |
| n |
| n | |
| \sum | |
| j=1 |
H(j)=
| infty | ||
| 1+\sum | \left(1- | |
| k=2 |
| 1 | |
| \zeta(k) |
\right)=1.705211...
where ΞΆ is the Riemann zeta function.
In the same paper Niven also proved that
| n | |
| \sum | |
| j=1 |
h(j)=n+c\sqrt{n}+o(\sqrt{n})
where h(1) = 1, h(n) = the smallest exponent appearing in the unique prime factorization of each natural number n > 1, o is little o notation, and the constant c is given by
c=
| |||||
| \zeta(3) |
,
and consequently that
\limn\toinfty
| 1 | |
| n |
| n | |
| \sum | |
| j=1 |
h(j)=1.