Niven's constant

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



\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^n H(j) = 1+\sum_{k=2}^\infty \left(1-\frac{1}{\zeta(k)}\right) = 1.705211\dots $$

where ζ is the Riemann zeta function.

In the same paper Niven also proved that



\sum_{j=1}^n 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 = \frac{\zeta(\frac{3}{2})}{\zeta(3)}, $$

and consequently that


 * $$ \lim_{n\to\infty} \frac{1}{n}\sum_{j=1}^n h(j) = 1. $$