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

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

where ζ is the Riemann zeta function.^[1]

In the same paper Niven also proved that

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

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

${\displaystyle c={\frac {\zeta ({\frac {3}{2}})}{\zeta (3)}},}$

and consequently that

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