content_image/340225.jpg content_image/340224.jpg content_image/340223.jpg content_image/340222.jpg content_image/340221.jpg
Lemma IV.2. For q = n n +2 we have 

<img src='content_image/340225.jpg'>

meaning that the quotient between the left- and right-hand side goes to 1 as n → ∞ . The proof follows from an easy application of the identity 

and we leave it to the reader. A somewhat more involved application of (35) is the following lemma 

Lemma IV.3. Let x be some real number. For q = n n +2 we have 

This allows us to give the exact order of the sum. 

Theorem IV.4. For q = n n +2 we have 

<img src='content_image/340224.jpg'>

<img src='content_image/340223.jpg'>

<img src='content_image/340222.jpg'>

Proof. The calculations goes as follows though we leave out some details. 

<img src='content_image/340221.jpg'>

where the factor n 3 / 4 in front of the integral comes from the change of variables k = x n 3 / $^{4}$. 

This is the sum at just a single point where the distribution becomes flat in the middle region. We can do even better if we allow ourselves to move around in the vicinity of this point. 

