Tagged Questions
5
votes
1answer
105 views
Interesting phenomenon with the $\zeta(3)$ series
I noticed that if one takes certain partial sums of the series for $\zeta(3)$:
$$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$
an interesting phenomenon occurs ...
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 ...
8
votes
1answer
272 views
Riemann's thinking on symmetrizing the zeta functional equation
In the translated version of Riemann's classic On the Number of Prime Numbers less than a Given Quantity, he quickly derives the zeta functional equation through contour integration essentially as
...
4
votes
1answer
422 views
Riemann zeta function and uniform convergence
A question in a past paper says prove that this series converges pointwise but not uniformly $$\xi(x):= \sum_{n=1}^\infty \frac{1}{n^x} .$$ But I thought that it did converge uniformly to some ...
13
votes
3answers
709 views
Computing $\zeta(6)=\sum\limits_{k=1}^\infty \frac1{k^6}$ with Fourier series.
Let $ f$ be a function such that $ f\in C_{2\pi}^{0}(\mathbb{R},\mathbb{R}) $ (f is $2\pi$-periodic) such that $ \forall x \in [0,\pi]$: $$f(x)=x(\pi-x)$$
Computing the Fourier series of $f$ and ...
2
votes
1answer
276 views
Primes and Riemann zeta function.
Primes numbers and Riemann zeta function.
Question 1: Is there a proof of the infinitude of prime numbers using the Riemann Zeta function. Exboço could show me a proof of this where I could find it?
...
1
vote
1answer
116 views
Can you provide a lower bound on $|\zeta (s) |$ for fixed $\mathrm{Re}(s) > 1$?
It's easy to prove, for example, that $|\zeta(2 + it)| > 2 - \frac{\pi^2}{6}$. However, there is some $\sigma > 1$ for which $\zeta ( \sigma ) = 2$, and it is more difficult to obtain a lower ...
34
votes
5answers
3k views
Why does $1+2+3+\dots = {-1\over 12}$?
$$\lim_{N\to\infty} \sum_{i=1}^N \, i = +\infty$$
$\displaystyle\sum_{n=1}^\infty n^{-s}$ only converges to $\zeta(s)$ if $\text{Re}(s) > 1$.
Why should analytically continuing to $\zeta(-1)$ ...
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
387 views
An Identity Concerning the Riemann Zeta Function
Let $\zeta$ be the Riemann- Zeta function. For any integer, $n \geq 2$, how to prove $$\zeta(2) \zeta(2n-2) + \zeta(4)\zeta(2n-4) + \cdots + \zeta(2n-2)\zeta(2) = \Bigl(n + ...
