n! ARITHMETIC FUNCTIONS

2 02 2009

Theorem:
If p_j is prime, then \displaystyle \alpha_j(n)=\sum_{i=1}^\infty \bigg\lfloor\frac{n}{p_j^i}\bigg\rfloor is the exponent of p_j appearing in the prime factorization of n!

Proof: (see [1] page 104, and [3])

But the infinity summation is applied to a infinity of zeros, because when \displaystyle \frac{n}{p_j^i}<1 , the terms vanish.

So the bigger exponent, m_j , that makes the terms not null is:
\displaystyle \\n{<}p_j^{m_j}; \rightarrow \frac{\log{n}}{\log{p_j}}{<}m_j; \rightarrow m_j=\bigg\lfloor\frac{\log{n}}{\log{p_j}}\bigg\rfloor; (m_j\in\mathbb{N})

\displaystyle \alpha_{j}(n)=\sum_{i=1}^{\big\lfloor\frac{\log{n}}{\log{p_j}}\big\rfloor}{\bigg\lfloor\frac{n}{p_j^i}\bigg\rfloor};

Now we can calculate any arithmetic function, like, e.g., the divisor sigma:

\displaystyle \sigma_{0}(n!)=\prod_{j=1}^{\pi(n)} (\alpha_j(n)+1)


References:


[1]-Number Theory George E. Andrews – Courier Dover Publications, 1994 – ISBN 0486682528, 9780486682525
[2]-ASYMPTOTIC ORDER OF THE SQUARE-FREE PART OF N! Kevin A. Broughan – Department of Mathematics, University of Waikato, Hamilton, New Zealand
[3]-FUNDAMENTOS DE LA TEORÍA DE LOS NÚMEROS – I. Vinogradov – Editorial MIR

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