Let and let be an -th primitive root of unity. Prove that
where denotes the Euler’s totient function.
where since is the minimal polynomial of over .
The result follows.
Let denote Euler’s totient function. Prove that for it holds that:
where stands for the Riemann zeta function.
Well by Euler’s product we have,
Combining we get the result.
Note: It also holds that
Consider the tridiagonal matrix . Its eigenvalues are . Hence,
and the result follows.
Let such that . Prove that
We are invoking the Hermite – Hadamard Inequality for the convex function . Thus,
Evaluate the integral
Successively we have:
We used the simple observation that