Questions on the various generalizations of the zeta function of Riemann. Consider using the tag (riemann-zeta) instead if your question is specifically about Riemann's function.
2
votes
1answer
200 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} ...
11
votes
1answer
228 views
Regularizing divergent series and Bernoulli numbers
Here is the actual problem I need a proof for: (I made an error writing down the equation initially)
$$B_{k+1} = \frac{(-1)^k}{2^{k+2}-2} \sum_{q=0}^{k} \binom{k+1}{q} 2^q B_q$$
Below is my ...
7
votes
1answer
193 views
Reference request: $L$-series and $\zeta$-functions
Does anyone know a good book, lecture note, article etc. on $L$-series (Dirichlet, Hecke, Artin) and $\zeta$-functions in number theory? I'm especially interested in material explaining the following:
...
8
votes
2answers
358 views
How to find $\zeta(0)=\frac{-1}{2}$ by definition?
I would like to know how we can find the following result:
$$\zeta(0)=-\frac12$$
Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$
to find this?
4
votes
0answers
180 views
New generalization of Riemann Zeta?
I am interested in the following generalization of the Riemann Zeta function:
$$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$
This is most closely related ...
5
votes
2answers
177 views
Lerch-$\small \zeta(\varphi,0,-n)$ of integer *n* purely real and imaginary ($\small \zeta_\varphi (-n)^2 $ is real) for $\small n \ge 2$?
Are the Lerch-$\zeta(\varphi,0,-n) $ of integer n (for shortness I use the notation of my earlier question $\small \zeta_\varphi(-n)$) periodically purely real and imaginary: $\zeta_\varphi (-n)^2 $ ...
2
votes
1answer
152 views
Periodic Zeta Function Functional Equation
Recall that the periodic zeta function has the Dirichlet series
$$F(\lambda,s)= \sum_{n=1}^\infty \frac{e^{2\pi i n\lambda}}{n^s}.$$
This defines an analytic function for $\Re s>0$ and has a ...
2
votes
4answers
346 views
Generalization $\zeta_\varphi(s)=\sum_{k=0}^\infty {\exp(I\varphi*k) \over (1+k)^s} $
This is more a reference-request for some fiddling/exploration with the $\zeta$-function. In expressing the $\zeta$ and the alternating $\zeta$ (="$\eta$") in terms of matrixoperations I asked myself, ...
4
votes
0answers
135 views
When functions, analytically continued, carry over certain properties
Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...
28
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 ...
5
votes
1answer
266 views
Square-free zeta function zeros
It is a well known fact that the geometric series
$$1+x+x^2+x^3+\ldots$$
has the following form
$$\frac{1}{1-x}$$
Another possible representation is
$$\prod_{k=0}^{\infty}\left(1+x^{2^{k}}\right)$$
...
2
votes
1answer
345 views
Is Riemann Zeta Function symmetrical about the real axis?
From wikipedia,
http://en.wikipedia.org/wiki/Riemann_zeta_function
"Furthermore, the fact that $\zeta(s) = \zeta(s^*)^*$ for all complex s ≠ 1 ($s^*$ indicating complex conjugation) implies that the ...
5
votes
1answer
272 views
Getting my number theoretic series straight
There are Artin $L$-series and Dirichlet $L$-series, and zeta functions for varieties and for number fields; there are a slew of objects named after Hecke... There are also various kinds of characters ...
4
votes
0answers
106 views
Shintani cone zeta function
Is there a procedure/algorithm for calculating sums of the form
$$ \sum_{n_1,\ldots,n_r >0} \frac1{L_1(n_1,\ldots,n_r)^{m_1} \ldots L_r(n_1,\ldots,n_r)^{m_r}} $$
where
$$ L_i(n_1,\ldots, n_r) ...
1
vote
2answers
127 views
On functions similar to Hurwitz zeta function
Denoted as $\zeta(s,a)$ for a > 0
Where do I find topics on the Hurwitz zeta function for a < 0?
Any links or resources would be appreciated.
(Please dont mention wiki or mathworld)
Thanks
3
votes
0answers
228 views
Connection between Bernoulli polynomials and polygamma function
There is an intricate connection between Hurwitz Zeta and the (traditional) polygamma function:
$$\psi_n(z)=(-1)^{n+1}n!\zeta(n+1,z)$$
If to use a generalization for Bernoulli numbers, this can be ...
2
votes
2answers
310 views
Some basic questions about the Selberg zeta function
I'm trying to learn about the Selberg zeta function, but it seems like introductory texts assume more knowledge of Riemannian geometry than I'm comfortable with.
I have some basic questions that ...
1
vote
2answers
178 views
Proving identity $\displaystyle\sum_{j\geq 1}[(j+t)^{-1}-j^{-1}]=\displaystyle\sum_{k\geq 1}\zeta (k+1)(-t)^{k}$
Motivation: In S.J. Patterson's An introduction to the theory of the Riemann
Zeta-Function it is proved (p.132) that
$\displaystyle -\Gamma ^{\prime }(t)/\Gamma (t)=\gamma +t^{-1}+\underset{j\geq ...
