Tagged Questions
3
votes
2answers
43 views
On the Hurwitz Zeta Function
In my mathematics course in Uni. (I'm a physics student) my prof. gave us the following exercise: to express the Hurwitz Zeta function $\zeta(2k+1,\frac{1}{4})$ with $k=1,2,3,\dots$ in terms of the ...
5
votes
1answer
100 views
Why $p$-adically interpolate?
I'm studying $p$-adic analysis now and particularly $p$-adic interpolation; for example, constructions like $p$-adic $L$-functions (Kubota-Leopoldt style). I'm having some difficulty though, and I'd ...
2
votes
1answer
120 views
Convergence of the Fourier Transform of the Prime $\zeta$ Functions
I think I found a way to write the truncated Prime $\zeta$ function like this:
$$
P_x(s)=\sum_{p<x} \frac{1}{p^s} =\sum_{n=1}^{\infty}\frac{ \mu (n)}{n}
\sum_{z\in\{1,\rho\}}(-1)^{1-\delta_{1z}}
...
1
vote
0answers
79 views
A Thue-Morse Zeta function ( Generalized Riemann Zeta function and new GRH )
Consider $t_n$ as the Thue-Morse sequence.
Let $m$ be a positive integer and $s$ a complex number.
Odiuos Number
Now consider the sequence of functions below
$f(1,s)=1+2^{-s}+3^{-s}+4^{-s}+...$
...
5
votes
1answer
111 views
On the Dirichlet beta function sum $\sum_{k=2}^\infty\Big[1-\beta(k) \Big]$
Given the Dirichlet beta function,
$$\beta(k) = \sum_{n=0}^\infty\frac{(-1)^n}{(2n+1)^k}$$
(The cases k = 2 is Catalan's constant.) It seems,
$$\sum_{k=2}^\infty\Big[1-\beta(k) \Big] = ...
2
votes
1answer
252 views
Why is $\pi$ the Limit of the Absolute Value of the Prime $\zeta$ Function?
Motivation:
I was looking at the approximation of the truncated Prime $\zeta$ function
$$
P_x(s)=\sum_{p\leq x}p^{-s}= \mathrm{li}(x^{1-s}) + O \left(\cdot \right)
$$
(to be found here with or ...
11
votes
1answer
216 views
What is the binomial sum $\sum_{n=1}^\infty \frac{1}{n^5\,\binom {2n}n}$ in terms of zeta functions?
We have the following evaluations:
$$\begin{aligned}
&\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\
&\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = ...
5
votes
0answers
119 views
Are there asymptotic expressions for multiple zetas $\small \zeta(s),\zeta(s,s),\zeta(s,s,s),\ldots$ where $\small s=1+\delta, \delta\to 0$?
Playing around with elementary symmetric functions I tried to generalize that to infinite series and arrived at the well known concept of MZV ("multiple zeta values"). At the moment I'm only ...
0
votes
1answer
173 views
What is the fractional derivative of the function $\pi \cot (\pi x)$?
What is the fractional derivative of the function $\pi \cot (\pi x)$?
I derived the following expression:
$(\pi \cot (\pi q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\gamma ) \zeta (p+1,q)}{\Gamma ...
3
votes
1answer
200 views
An example of divergent series with the Lerch function
I am often working with divergent series all around being this the bread and butter for a theoretical physicist. Thanks to the excellent work of Hardy these have lost their mystical Aurea and so, they ...
4
votes
0answers
175 views
New generalization of Riemann Zeta?
I am interested in the following generalization of the Riemann Zeta function:
$$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$
This is most closely related ...
5
votes
2answers
177 views
Lerch-$\small \zeta(\varphi,0,-n)$ of integer *n* purely real and imaginary ($\small \zeta_\varphi (-n)^2 $ is real) for $\small n \ge 2$?
Are the Lerch-$\zeta(\varphi,0,-n) $ of integer n (for shortness I use the notation of my earlier question $\small \zeta_\varphi(-n)$) periodically purely real and imaginary: $\zeta_\varphi (-n)^2 $ ...
2
votes
1answer
141 views
Periodic Zeta Function Functional Equation
Recall that the periodic zeta function has the Dirichlet series
$$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
326 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, ...
2
votes
1answer
334 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 ...
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
1
vote
2answers
177 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 ...
