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Calculation of the Mean

This section is devoted to finding the limit of the mean of Xn,a. The following theorem is the main result of this paper, and will be proved in sections 6.1 and 6.2.

Theorem 1   If a=0, then

\begin{displaymath}\lim_{n\rightarrow\infty}E[X_{n,a}] =
\ln \left(l^{\lfloor l \rfloor} \over \lfloor l \rfloor! \right),
\end{displaymath}

and if $a=\frac{p}{q}$ with p and q relatively prime, then

\begin{displaymath}\lim_{n\rightarrow\infty}E[X_{n,a}] = \frac{1}{q} \ln \left(\frac{(ql)^{\lfloor ql
\rfloor}}{\lfloor ql \rfloor !}\right).
\end{displaymath}

 

The proof is divided into two parts. The result is proved first for the case when a=0, and then is extended to include any rational a. Although the proof splits naturally in this way, the formula for a=0 actually corresponds to q=1 in the more general case. The case when a is irrational will be looked at in section 6.3.



 


2000-09-25