Niven's constant

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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

where ζ(k) is the value of the Riemann zeta function at the point k (Niven, 1969).

In the same paper Niven also proved that

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

and consequently that

References[edit]

  • Niven, Ivan M. (August 1969). "Averages of Exponents in Factoring Integers". Proceedings of the American Mathematical Society. 22 (2): 356–360. doi:10.2307/2037055. JSTOR 2037055.
  • Steven R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 2003

External links[edit]