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## Zeta logarithmic series

Let denote the zeta function. Prove that

**Solution**

## Mellin transform integral

Evaluate the integral:

**Solution**

We are evaluating the Mellin transform of the function .

where is the Euler’s Gamma function. Hence,

where is the Euler – Mascheroni constant and is the digamma.

## Contour integral

Let be analytic in the disk . Prove that:

**Solution**

It follows from Taylor that and the convergence is uniform. Hence,

We have that and for we also have that due to

So if then lies within the disk ; hence the integral equals whereas if then lies outside the disk ; hence

## Functions that preserve convergent serieses

Find all functions that preserve convergent serieses.

## Root inequality

Let be positive real numbers such that . Prove that

**Solution**

Due to the AM – GM we have that

(1)

and

(2)

Thus,