Niven's constant

The Niven 's constant, named after the Canadian-American mathematician Ivan M. Niven, is a mathematical constant in number theory. It is defined as the limit of the arithmetic mean of the maximum exponent of the first prime factorizations natural numbers for.

Definition

It should be an integer with prime factorization with and for, and also the maximum of the exponents in the prime factorization of ( sequence A051903 in OEIS ), for example, the numbers with just the square-free numbers. Thus Niven constant is defined as

Properties

The Niven 's constant can be expressed by the Riemann zeta function and in this way be calculated approximately ( Niven 1969):

For the asymptotic behavior of the minima of the exponents Niven proved at the suggestion of Erdős

Where and are the minimum of the exponents in the prime factorization of ( sequence A051904 in OEIS ) and a Landau symbol. Thus, in particular