2
votes
1answer
71 views

The multiplication formula for the Hurwitz/generalized Riemann zeta function

I'm having a difficult time showing that $$ \displaystyle \zeta(s,mz) = \frac{1}{m^{s}} \sum_{k=0}^{m-1} \zeta \left(s,z+\frac{k}{m} \right) $$ A couple of authors referred to it as an obvious fact. ...
53
votes
3answers
4k views

Evaluate $\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx$

I am trying to find a closed form for $$\int_0^1 \frac{\log \left( 1+x^{2+\sqrt{3}}\right)}{1+x}\mathrm dx = 0.094561677526995723016 \cdots$$ It seems that the answer is $$\frac{\pi^2}{12}\left( ...
2
votes
3answers
243 views

Riemann Zeta formula

can anyone check if this formula is plausible ?? $$ \frac{1}{\zeta (s)} = \sum_{n=0}^{\infty}\frac{ (-\pi)^{n}(s-1)s}{2n!(s+2n)(s+2n+1)} $$ according to the authors this formula would be valid only ...
2
votes
1answer
203 views

What is this series called and when does it diverge?

What is this series called (if it has a name)? When does it diverge without analytic continuation and when does it diverge with analytic continuation? $\sum_{k_1,\dots,k_m=1}^{\infty} ...
32
votes
5answers
2k views

Does $\zeta(3)$ have a connection with $\pi$?

The problem Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)? Details Several ...