# Tagged Questions

Questions on the famed $\zeta(s)$ function of Riemann, and its properties.

410 views

### Infinite Series $\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}$

How can I prove that $$\sum_{n=1}^\infty\frac{H_{2n+1}}{n^2}=\frac{11}{4}\zeta(3)+\zeta(2)+4\log(2)-4$$ I think this post can help me, but I'm not sure.
180 views

### Dirichlet series and Riemann zeta function

Im trying to show, for $\Re(s)>1$, that $\displaystyle\sum_{n=0}^{\infty} \frac{d(n^2)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)}$, where $d(n)= |\{k \mid k|n \}|$, number of positive integers that ...
100 views

49 views

108 views

130 views

### Series involving the Riemann zeta function

Consider the series: $$\sum_{n=1}^{\infty}\frac{\zeta(2n+1)}{n(2n+1)}$$ We can easily prove that it's a convergent series. My question, is there a way to express this series in terms of zeta ...
274 views

### An interesting identity involving powers of $\pi$ and values of $\eta(s)$

It happens that, for any $m\geq 1$, $$\sum_{n=0}^{+\infty}\frac{(-1)^n}{(2n+1)^{2m+1}}=\frac{E_{2m}}{2\cdot(2m)!}\left(\frac{\pi}{2}\right)^{2m+1}\tag{1}$$ where $E_{2m}$ is an integer number. My ...
91 views

### Ramanujan's divergent series

I tried to prove this sum by myself, but I couldn't. $1 + 4 + 9 + 16 + ... = 0$ First, I know this sums are a bit problematic, as we can't just $'='$ an infinite sum, but I would like to see the ...
51 views

### What's about the derivative of the Riemann zeta function?

The derivative of the Riemann Zeta function is $$\zeta'(s)=-\sum_{n=2}^\infty\frac{\log n}{n^s}$$ for $\Re s>1$. Question. Can you refers us in a short post, from a divulgative viewpoint (but ...
191 views