Pseudo sum

Let $\alpha, \beta $ be positive irrational numbers such that $\displaystyle \frac{1}{\alpha} + \frac{1}{\beta}=1$. Evaluate the (pseudo) sum:


$$\sum_{n=1}^{\infty} \left(\frac{1}{\lfloor n\alpha\rfloor^2}+\frac{1}{\lfloor n\beta\rfloor^2}\right)$$

Solution

Alternating series with eta dirichlet

Let $\eta$ denote Dirichlet's eta function. Prove that

$$\frac{\pi}{4}=1+\sum_{k=1}^{\infty}(-1)^{k}\frac{\eta(k)}{2^{k}}$$

Solution

$\sum \limits_{n=1}^{\infty} \left(\zeta(2n) -\beta(2n) \right)$

Prove that:

$$\sum_{n=1}^{\infty}\left( \zeta(2n)-\beta(2n) \right)=\frac{1}{2}+\frac{\ln 2}{2}$$

Solution

A $\zeta(2n+1)$ series

Let $\zeta$ denote the Riemann Zeta function. Evaluate the series:

$$\sum_{n=1}^{\infty} \frac{\zeta(2n+1)}{(n+1)(2n+1)}$$

(Serafim Tsipelis, Anastasios Kotronis)

Solution

Diriclet series

Let $\sigma(n)$ be the divisor function that is $ \displaystyle {\rm \sigma(n)=\sum \limits_{d \mid n} d}$. Prove that for $s \in \mathbb{R} \mid s >2$ it holds that:

$$\sum_{n=1}^{\infty} \frac{\sigma(n)}{n^s} = \zeta(s) \zeta(s-1)$$

where $\zeta$ is the Riemann zeta function.

Solution

Series

Let $a>\frac{1}{4}$. Prove that:

$$\sum_{n=1}^{\infty} \frac{1}{n^2-n+a} = \frac{\pi}{\sqrt{4a-1}} \frac{e^{\pi \sqrt{4a-1}}-1}{e^{\pi \sqrt{4a-1}}+1}$$

Solution