Niven's constant
Niven's constant is a number in number theory named after Ivan Niven. It describes the typical size of the largest exponent that appears in the prime factorization of natural numbers, when you look at many numbers and average.
Definition. For any positive integer n, write n as a product of primes with exponents: n = p1^a1 p2^a2 ... pk^ak, where each ai ≥ 1. Let H(n) be the largest exponent among the ai. We set H(1) = 1. Niven showed that the average value of H(n) as n grows large settles to a fixed constant. This constant can be expressed by a series involving the Riemann zeta function.
Niven also proved a related result for the smallest exponent in the factorization. If h(n) denotes the smallest exponent (again with h(1) = 1), there is a parallel limiting constant described similarly, together with an associated error term described by little-o notation. The precise formulas tie these constants to infinite series built from the zeta function.
This page was last edited on 2 February 2026, at 07:53 (CET).