## Galois theory … of the Euler’s totient function

Let and let be an -th primitive root of unity. Prove that

where denotes the Euler’s totient function.

Solution

We have

where since is the minimal polynomial of over .

The result follows.

## On Euler’s totient function series

Let denote Euler’s totient function. Prove that for it holds that:

where stands for the Riemann zeta function.

Solution

Well by Euler’s product we have,

thus,

(1)

and

(2)

Combining we get the result.

Note: It also holds that

## Trigonometric sum

Prove that

Solution

Consider the tridiagonal matrix . Its eigenvalues are . Hence,

and the result follows.

## Logarithmic mean inequality

Let such that . Prove that

Solution

We are invoking the Hermite – Hadamard Inequality for the convex function . Thus,

## On an integral

Evaluate the integral

Solution

Successively we have:

We used the simple observation that