Calculation of the Mean

This section is devoted to finding the limit of the mean of X_n,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

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

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

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

 

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.