80 votes
Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?
The intuition may be helped by considering the generalized Euler constant function $$\gamma(z)=\sum_{n=1}^\infty z^{n-1}\left(\frac{1}{n}-\ln\frac{n+1}{n}\right),\;\;|z|\leq 1.$$ Its values include ...
- 201k
37 votes
Accepted
Is the integral of $e^{-x^2}$ from $0$ to $1$ known to be irrational?
Following @te4's comment, we can look at the function $$f(x) = \frac{\sqrt{\pi}}{2\sqrt x} \operatorname{erf}(\sqrt{x}) = \sum_{n=0}^\infty \frac{(-1)^n x^n}{(2n+1) n!}.$$ Note that it's an E-function,...
- 4,868
32 votes
Accepted
What is this modified arithmetico-geometric mean function?
The function $A(x,y)$ has the closed-form expression $$A(x,y)=\frac{\sqrt{y^2-x^2}}{\arccos(x/y)}.$$ This modification of the AGM was introduced by Gauss in 1800, in an unpublished letter. It was ...
- 201k
27 votes
Accepted
25 votes
Accepted
Representations of $\zeta(3)$ as continued fractions involving cubic polynomials
See NOTE below. This MO inquiry is over 3 yrs old now. By the date the question about the $\zeta(3)$ CF with $k=8/7$ was made (Feb, 2019), it can be answered in the negative nowadays, since it was '(...
- 3,271
23 votes
Is the integral of $e^{-x^2}$ from $0$ to $1$ known to be irrational?
$\newcommand{\Q}{\mathbb Q}\newcommand{\erf}{\operatorname{erf}}$(This answer had been posted before I saw Command Master's answer. I am leaving it here, since it contains more and/or different ...
- 143k
22 votes
A new way of approaching the pole of the Riemann zeta function - and a new conjectured formula
The paper entitled Euler constant as a renormalized value of Riemann zeta function at its pole by Andrei Vieru contains a derivation of the first formula in the OP (Benoȋt Cloitre's formula), and ...
- 201k
22 votes
Closed form of an infinite series
Denote $c_n:={(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin(\frac{2\pi n}{3})}$ the $n$-th term of the series. We have for all $k\ge0$ $$c_{3k}=0,$$ $$c_{3k+1}= (-1)^{k+1}\...
- 63.5k
18 votes
Accepted
Is there a specific named function that is the inverse of $x+x^a$ for $x$ real?
The answer is yes indeed. It is a special case of Fox-H function, a variation of the confluent Fox-Wright $_{1}\Psi_{1}$ function (a generalization of the confluent hypergeometric function $_{1}F_{1}$)...
- 3,271
18 votes
Power series $x^n/n$ with one plus, then two minuses, then three plusses, and so on
The answer is in terms of theta functions. Let $$\Psi(q)=1+q+q^3+q^6+q^{10}+q^{15}+\cdots;$$ this is a Jacobi theta function that can be written as $q^{-1/8}\eta^2(2\tau)/\eta(\tau)$ where $q=e^{2\pi ...
- 22.4k
17 votes
Accepted
Any name for this special function?
This is a standard hypergeometric function. Note that $$ \frac{1}{(a-m)!} = (-1)^m \frac{(-a)_m}{a!}\quad\text{and}\quad \frac{1}{(b+m)!} = \frac{1}{b!\,(b+1)_m}$$ in terms of the rising Pochhammer ...
- 4,002
17 votes
Why does this theta function value yield such a good Riemann sum approximation?
This is because the theta function has a functional equation. Under the usual definition $\theta(t) = \sum_n e^{-\pi t n^2}$, the functional equation is $$\theta(t) = \frac 1{\sqrt t} \theta\left(\...
- 3,472
16 votes
A new way of approaching the pole of the Riemann zeta function - and a new conjectured formula
The first formula is trivial. $$f(s)= \frac1{s-1}+\gamma +O(s-1)$$ $$g(z)=1+2^{-z}+3^{-z}+4^{-z}+O(5^{-z})=1+2^{-z}(1+(3/2)^{-z}+(4/2)^{-z}+O(5/2)^{-z})$$ $$f(g(z)) = \frac1{2^{-z}(1+(3/2)^{-z}+(4/2)^...
- 3,434
16 votes
Accepted
Question about functions $f: \mathbb{Z}^+ \to \mathbb{Z}^+$ such that $x$ is prime whenever $f(x)$ is prime
As observed in comments, we have $f(n) = \lfloor g(n) \rfloor$ where $g(n) = \frac{\alpha^n + \alpha^{-n}}{4}$ and $\alpha = 2 + \sqrt{3}$. From the recurrence $g(n+1) = 4 g(n) - g(n-1)$ we see that $...
- 120k
15 votes
Accepted
Short research articles
[a bit too long for a comment] I understand from the question that the aim is to find a research project based on the search for a counterexample. By construction, this will mean showing that some ...
15 votes
Accepted
Origins of the Bessel function (particularly of the 1st kind)
The story is explained in detail in the first chapter of the book: G. N. Watson, Treatise on the theory of Bessel functions, Cambridge, 1922. A Riccati equation $y'=y^2+x^2$ first appears in the work ...
- 96.7k
14 votes
Accepted
Several conjectured identities for polylogarithms
(Updated answer): Upon further research, it turns out your three equations involving $\phi$ are special cases of three polylogarithm ladders of index $12,\,20,\,24$ that can be found in "The ...
- 13k
14 votes
Accepted
Series involving factorials
The sum $$\sum_{k=0}^\infty \frac{(a+k)!\,(b+k)!}{k!\,(a+b+c+k+1)!}z^k.$$ is not only a generalized hypergeometric series; it's the original ungeneralized Gauss hypergeometric series, $$\frac{\Gamma(a+...
- 17.7k
14 votes
Accepted
A surprising identity: $\det[\cos\pi\frac{jk}n]_{1\le j,k\le n}=(-1)^{\lfloor\frac{n+1}2\rfloor}(n/2)^{(n-1)/2}$
First of all, we use the formula $$ D:=\det [x_j^k+x_j^{-k}]_{j,k=0,\dots,m-1}=\prod_{l<j}(x_j+x_j^{-1}-x_l-x_l^{-1})=\prod_{l<j} (x_j-x_l)(1-x_j^{-1}x_l^{-1}). $$ This follows from the ...
- 115k
14 votes
A new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$?
If by "the class number $h(p^*)$ of the quadratic field $\mathbb{Q}(\sqrt{p^*})$" you mean "the minus class number $h^{-}$ of $\mathbf{Q}(\zeta_p)$" and if by " a possible new formula for the ...
- 165
14 votes
Accepted
A multiple integral that seems related to the $\zeta$ function at even integers
We have $$I_{2n}=\frac{(2n)!!}{(2n+1)!!}\cdot \frac1{2n+2}\cdot \pi^{2n}.$$ To see this, we follow the suggestion by Terry Tao in the comments and apply the diagonalization of the integral operator ...
- 115k
14 votes
Accepted
Why does this theta function value yield such a good Riemann sum approximation?
I would give a slightly more direct explanation than WhatsUp. The Poisson summation formula states that for $f$ a smooth rapidly decreasing function on $\mathbb R$, $\hat{f}$ the Fourier transform, ...
- 162k
13 votes
A new formula for the class number of the quadratic field $\mathbb Q(\sqrt{(-1)^{(p-1)/2}p})$?
The conjecture is not true, as some examples show. Let $D(p)$ denote your number. For primes $p \equiv 1 \textrm{ mod } 4$, we have $D(29)=8$, $D(37)=37$, $D(41)=121$ while $h(29)=h(37)=h(41)=1$. ...
- 21.1k
13 votes
Accepted
Speed of convergence of $\zeta(2k)\to 1$?
Here is an explicit bound. The sum $\sum_{n > N} n^{-s}$ for real $s > 1$ is bounded by the integral $$\int_N^\infty x^{-s} = N^{1-s} / (s-1).$$ Therefore for any $N$ you have $$0 < \zeta(s) -...
- 11.5k
13 votes
Accepted
Is the Gauss hypergeometric series ${}_2F_1\bigl(\frac{1}{2},\frac{1}{2};2;t\bigr)$ an elementary function?
Maple does it in terms of complete elliptic integrals $\rm{K}$ and $\rm{E}$ ... $$ {\mbox{$_2$F$_1$}\left(\frac12,\frac12;\,2;\,t\right)}={\frac {4\left( t-1 \right){\rm K} \left( \sqrt {t} \right) ...
- 41.7k
13 votes
Accepted
Riemann, fluid dynamics, and critical lines
Q: Does anyone know of a reference which discusses more thoroughly the critical line appearing in Riemann's hydrodynamics problem? A: A recent reference is Elliptical instability in hot Jupiter ...
- 201k
13 votes
Accepted
Conjectured closed form of $\int_0^1 \frac{\text{Li}_2\left(\frac{x}{4}\right)}{4-x}\,\log\left(\frac{1+\sqrt{1-x}}{1-\sqrt{1-x}}\right)\,dx$
As Henri Cohen remarked, the identity to prove is equivalent to $$\sum_{n=1}^\infty \frac{H_n^{(2)}}{(n+1)(2n+1)\binom{2n}{n}}=\frac{\pi^4}{972}.\tag{1}$$ In turn, this follows readily from the OP's ...
- 114k
13 votes
Accepted
Missing factor of 10 in derivation for integral form of ζ(3)
It is $-5\zeta(3)$. Moreover, we have separately (1) $\int \frac{\log(1-xy)}{xy}=-\zeta(3)$ and (2) $\int \frac{\log(1-\sqrt{xy})}{xy}=-4\zeta(3)$. For proving (1), expand the log as a Taylor series ...
- 115k
13 votes
Accepted
Proving identity involving dilogarithms and $\pi/9$
1. I believe that the first (and perhaps only) published proof of this identity appears in Kirillov's 1995 paper Dilogarithm identities. This paper was also pointed out by Carlo Beenakker in his (...
- 114k
12 votes
Accepted
Evaluating elliptic integrals
This question has been out for a while, and I think it deserves a thorough answer, even tho I came quite late to this party. As both the classical Legendre-Jacobi theory and the Carlson theory have ...
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