Let . Evaluate the integral
The key substitution is . Applying it we see that
Thus , .
Let . Prove that .
The characteristic polynomial of is . This in return means and . Thus,
Converting the product to a sum and using duplication formula for the gamma function and telescoping,
Using Stirling formula
we get that
Let be the roots of the polynomial
Suppose the roots of polynomial are where
Let . Then, the are the zeros of in the disk where is chosen such that for .
Jensen’s inequality implies that
Applying Cauchy – Schwartz yields,
Letting and we get the result.
Let denote the – th harmonic number. Evaluate the limit
Lemma: It holds that .
Proof: We have successively: