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# Tag Archives: Special Functions

## Double summation

Evaluate the sum

Solution

Lemma: It holds that

Proof: Consider the function and let us it integrate over the following contour

By the residue theorem it follows that

For the residues we have

The integrals along the sides vanish; hence:

and since

the result follows.

Back to the problem. We have successively:

## Arithmotheoretic sum

Evaluate the sum

Solution

The sum converges absolutely , so we can switch the order of summation; hence:

The last sum equals and hence

## Gamma inequality

Let be three positive real numbers such that . Prove that:

where is Euler’s Gamma function.

Solution

We can rewrite the inequality as

since is convex.

## Asymptotic expansion

Prove that

for some constant .

## A product

Let denote the golden ratio , the Möbius function and   Euler’s totient function. Prove that:

Solution

We are using the following facts:

(1)

(2)

Hence,