number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, Niven's constant, named after Ivan Niven, is the largest exponent appearing in the

prime factorization In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization. When the numbers are suf ...

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_ \frac \sum_^n H(j) = 1+\sum_^\infty \left(1-\frac\right) 
= 1.705211\dots

where ζ is the

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...

. In the same paper Niven also proved that :

\sum_^n h(j) = n + c\sqrt + o (\sqrt)

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,

and consequently that :

\lim_ \frac\sum_^n h(j) = 1.

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External links

* * Mathematical constants Number theory {{Numtheory-stub

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