Niven's constant

Template:Short description Template:Use shortened footnotes 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

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

where ζ is the Riemann zeta function.Template:R

In the same paper Niven also proved that

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

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

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

and consequently that

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

Further reading

• Steven R. Finch, Mathematical Constants (Encyclopedia of Mathematics and its Applications), Cambridge University Press, 2003

External links

• Template:MathWorld
• Template:OEIS el

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