Start with the following manipulations:
The expression is just a Gauss sum, and has magnitude . Hence
Since is odd, we have
Now for ; to prove this, it suffices to show that the function given by is decreasing and approaches 1 as . To prove the latter statement, substitute and take the limit as using L’Hôpital’s rule. To prove the former statement, it will suffice to show that is less than zero on the interval . But as and is increasing on , since for , so is less than zero for .
With this in hand, we have
|Date of creation||2013-03-22 12:46:23|
|Last modified on||2013-03-22 12:46:23|
|Last modified by||djao (24)|