All Questions
11 questions
71
votes
8
answers
12k
views
Possible new series for $\pi$
In a recent (unfortunately over-hyped) preprint by Saha and Sinha, Field theory expansions of string theory amplitudes (arXiv:2401.05733), they present the following series for $\pi$:
$$\pi = 4 + \...
- 82.7k
16
votes
1
answer
4k
views
Order of magnitude of $\sum \frac{1}{\log{p}}$
Question: What is the order of magnitude of the following sum?
$$ \sum_{\substack{p<n\\\text{$p$ prime}}} \frac{1}{\log{p}} $$
Additional information: Since
$$ \sum_{\substack{p<n\\\text{...
- 3,004
10
votes
0
answers
350
views
How are the hypergeometric motives of WZ-Pairs connected?
If $\small{(F,G)}$ is a WZ-pair and general asymptotic conditions $\lim_{k\rightarrow\infty}\small{G(n,k)=0}$ and $\lim_{n\rightarrow\infty}\small{F(n,k)=0}$ hold, then we have the certified ...
- 2,836
10
votes
1
answer
554
views
Who was/were the first to note that if $\sum_{x \in X} \frac{1}{x} < \infty$ then the natural density of $X$ is zero?
It is a result of folklore that the natural density of a set $X$ of positive integers such that $\sum_{x \in X} \frac{1}{x} < \infty$ is zero. This is reproved, e.g., in T. Šalát's paper: ...
- 10.5k
42
votes
4
answers
4k
views
Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?
Now that some of the previously MSE formulae that I left here have been applied Dec.2023 to compute high precision record values ($10^{12}$ decimal digits) of trascendental constants $\Gamma(1/3)$ (Eq....
- 2,836
15
votes
3
answers
1k
views
Does anyone remember what happened to the experimental search for polynomial identities for $\pi$?
So a while back I was on the internet and had encountered a website containing an experimental search for identities for $\pi$. My memory was that the page belonged to either Jonathan Sondow or ...
- 2,689
13
votes
1
answer
1k
views
Apéry's constant $\zeta(3)$ fastest convergent series
UPDATE Feb.02.2024
The series below, Eq.(3) for computing and Eq.(2) for verifying, were applied by Andrew Sun on Dec.22.2023 to get over $2\cdot10^{12}$ decimal digits and break the number of ...
- 2,836
5
votes
1
answer
819
views
Is this known alternating sum for Euler's constant?
This probably is known, but Wolfram Alpha doesn't recognize it
and couldn't find it in Mathworld (there is something close,
but using floor).
We have
$\lim_{s \to 1} (\zeta(s)-1/(s-1)) = \gamma$
...
- 25.4k
5
votes
1
answer
613
views
generating $q$-Catalan numbers
An $n$-Dyck path (or a Catalan path) is a lattice path $P$, unit East and North steps, in an $n\times n$ square grid which stays (weakly) above the main diagonal. Let $\square_n$ denote all such paths....
- 43.2k
5
votes
3
answers
300
views
Closed formula for $(-1)$-Baxter sequences
The number of the so-called Baxter permutations of length $n$ is computed by
$$a_n=\frac1{\binom{n+1}1\binom{n+1}2}\sum_{k=0}^{n-1}\binom{n+1}k\binom{n+1}{k+1}\binom{n+1}{k+2}.$$
There has also been a ...
- 43.2k
1
vote
1
answer
344
views
Products involving exponents of tribonacci numbers
The Fibonacci numbers $F_n$ can be given by
$$\sum_{k\geq0}F_kx^k=\frac{x}{1-x-x^2}.$$
Among many many properties of this sequence, consider the following two results:
(1) the coefficients of the ...
- 43.2k