Tagged Questions
4
votes
3answers
168 views
Factorial of infinity
So, I've read in this article that:
$$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$
And that:
$$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$
I found this result very ...
6
votes
1answer
140 views
Apéry's constant ($\zeta(3)$) value
I tried to find some proofs about the Apéry's constant, but I didn't find any intuitive proof. Is this constant given by the "brutal force" summing of $1 + \frac{1}{2^3} + \frac{1}{3^3} + ...
0
votes
1answer
56 views
Reformulation of riemann zeta
Does this extend to $\mathbb{C}$?
$\displaystyle ζ(x) = \int_0^{\infty} \frac{ 1}{\lfloor t\rfloor ^x} dt$, where for $0 \leq t < 1$ we say that $\lfloor t \rfloor = 1$.
7
votes
1answer
90 views
Serre's proof that zeta function is meromorphic
I try to understand the proof of Chap. VI, n° 3.1, Prop. 10 in Serre's "A course in arithmetic" (page 70). The goal is to prove that zeta-function can be written as
\begin{align*}
...
1
vote
2answers
74 views
Estimating the integrated Tchebychev function and calculating its error
I would like to understand how to derive (2) from (1) below.
Problem:
If $\psi_1$ is the integrated Tchebychev function below
$$\psi_1(x)=\frac{1}{2\pi i} ...
1
vote
0answers
92 views
How to get the floor function as a Mellin inverse of the Hadamard product of the Riemann zeta function?
The floor function is given - by Perron's formula - as a Mellin inverse of the zeta function. namely :
$$\left \lfloor x \right \rfloor=\frac{1}{2\pi ...
6
votes
1answer
127 views
Zeta functions of groups and an identity of Ramanujan
Zeta functions are being developed for all sorts of mathematical objects these days. One general situation is that of zeta functions of groups. If $G$ is a finitely generated group then we let
...
2
votes
3answers
192 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 ...
3
votes
0answers
124 views
Is this formula for $\zeta(15)$ true?
Apery gave,
$\begin{aligned}
\zeta(3) &= \frac{5}{2}\,\sum_{k=1}^\infty \frac{(-1)^{k+1}}{k^3\,\binom {2k}k}\end{aligned}$
J. Borwein and D. Bradley found this can be generalized to ...
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:
$$\begin{aligned}
&\sum_{n=1}^\infty \frac{1}{n\,\binom {2n}n} = \frac{\pi}{3\sqrt{3}}\\
&\sum_{n=1}^\infty \frac{1}{n^2\,\binom {2n}n} = ...
7
votes
2answers
330 views
How to find $\zeta(0)=\frac{-1}{2}$ by definition?
I would like to know how we can find the following result:
$\zeta(0)=\frac{-1}{2}$
Is there a way, using the definition, $$\zeta(s)=\sum_{i=1}^{\infty}i^{-s}$$
to find this?
4
votes
0answers
175 views
New generalization of Riemann Zeta?
I am interested in the following generalization of the Riemann Zeta function:
$$ \zeta_M(s,c) = \sum_{n=1}^\infty \left(\frac{n^2}{c^2} + \frac{c^2}{n^2}\right)^{-s} $$
This is most closely related ...
28
votes
5answers
2k views
Does $\zeta(3)$ have a connection with $\pi$?
The problem
Can be $\zeta(3)$ written as $\alpha\pi^\beta$, where ($\alpha,\beta \in \mathbb{C}$), $\beta \ne 0$ and $\alpha$ doesn't depend of $\pi$ (like $\sqrt2$, for example)?
Details
Several ...
5
votes
1answer
262 views
Square-free zeta function zeros
It is a well known fact that the geometric series
$$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)$$
...
