Series with cyclotomic polynomials
Posted: Tue Sep 13, 2016 10:02 pm
Let $\Phi$ denote the $n$ -th cyclotomic polynomial. Prove that:
$$\sum_{n=1}^\infty\frac{1}{n^4}\frac{\Phi'_n(e^{2\pi})}{\Phi_n(e^{2\pi})}=\frac{45\zeta(3)}{\pi^4e^{2\pi}}+\frac{7}{4\pi e^{2\pi}}$$
where $\Phi'$ stands for the derivative.
$$\sum_{n=1}^\infty\frac{1}{n^4}\frac{\Phi'_n(e^{2\pi})}{\Phi_n(e^{2\pi})}=\frac{45\zeta(3)}{\pi^4e^{2\pi}}+\frac{7}{4\pi e^{2\pi}}$$
where $\Phi'$ stands for the derivative.