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77 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 ...
Carlo Beenakker's user avatar
33 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,...
Command Master's user avatar
27 votes
Accepted

What is the value of $j(2\sqrt{-163})$?

The minimal polynomial is ...
Noam D. Elkies's user avatar
24 votes
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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 '(...
Jorge Zuniga's user avatar
  • 2,210
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 ...
Carlo Beenakker's user avatar
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}\...
Pietro Majer's user avatar
  • 56.6k
21 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 ...
Iosif Pinelis's user avatar
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 ...
Dave Benson's user avatar
  • 11.9k
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 ...
Timothy Budd's user avatar
  • 3,555
16 votes
Accepted

are these polynomials or rationals functions?

This is response to QUESTION 1. As Fedor pointed out, we're dealing with the Chebyshev polynomials $P_n(2\cos t)=\sin nt/\sin t$. So we must show that if $$ \sum_{n=1}^N \sin nt = 0 , \quad\quad\quad\...
Christian Remling's user avatar
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)^...
reuns's user avatar
  • 3,405
16 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 $_{1}F_{1}$)...
Jorge Zuniga's user avatar
  • 2,210
15 votes

How to prove Lambert's W function is not elementary?

In [Ritt 1948] p. 53 - 56, the method of J. Liouville is given for Kepler's equation. The same method can be applied to functions $f$ with $f(z)=A(z,e^z)$ ($A$ an algebraic function of two complex ...
IV_'s user avatar
  • 1,063
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

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 $...
Terry Tao's user avatar
  • 109k
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+...
Ira Gessel's user avatar
  • 16.2k
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 ...
Fedor Petrov's user avatar
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 ...
user134696's user avatar
13 votes
Accepted

Summation of series involving $\sinh$ of a square root

Here is a solution that I have found while working on other lattice sums. It utilizes a very simple result: Define $f$ by $$f(x)=\sum_{n=0}^{\infty} (-1)^n (2n+1) e^{-\pi x (n+\frac12)^2}.$$ ...
Noam Shalev - nospoon's user avatar
13 votes
Accepted

Rotation invariance of an integral

Yes. Denote $p=\sinh s$, then $s\in (-\infty,\infty)$, $\sqrt{p^2+1}=\cosh s$, $dp=\cosh s\, ds$. Next, denote $r=\sqrt{\tau^2+x^2}$, $\tau=r\cos \varphi$, $x=r\sin \varphi$, where $\varphi\in (-\pi/2,...
Fedor Petrov's user avatar
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 $h(29)=h(37)=h(41)=1$. ...
François Brunault's user avatar
13 votes
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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 ...
Fedor Petrov's user avatar
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) -...
Sean Eberhard's user avatar
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 ...
Carlo Beenakker's user avatar
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 ...
J. M. isn't a mathematician's user avatar
12 votes
Accepted

Asymptotics of a special function

Gradshteyn & Ryzhik equation 3.773.1 gives (for $q>0$) $$\frac{1}{q}B(q)=\int_0^\infty \frac{\sin x}{x(q^2/4+x^2)^{1/2}}\,dx=\frac{1}{q}G^{21}_{13}\left(\frac{q^2}{16}\biggl|^1_{1/2,1/2,0}\...
Carlo Beenakker's user avatar
12 votes

A tantalizing Gamma quotient to challenge the Rohrlich-Lang Conjecture

A conceptual way to tackle this question is to look at universal distributions on $\mathbf{Q}/\mathbf{Z}$, studied by Kubert and Lang among others. Distributions arise naturally in number theory, see ...
François Brunault's user avatar
12 votes

A variant of Lambert function

Sorry, my answer is wrong. As it was pointed out by "Simply Beautiful Art" $x\ne W(z)$ but $x=e^{W(z)}.$ It is known that $$W'(z)=\frac{W(z)}{z(1+W(z))}.$$ Let $z=\log t$. Then $x=W(z)$ is a root of ...
Alexey Ustinov's user avatar
12 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) ...
Gerald Edgar's user avatar
  • 40.3k
12 votes
Accepted

Closed form of an infinite series

Q: Does the following infinite series have a closed form? It does, according to Mathematica: $$\sum_{n=1}^{\infty} {(-1)^n \frac{\Gamma(\frac{1}{3}+\frac{n}{3})}{\Gamma(1+\frac{n}{3})} \sin\left(\frac{...
Carlo Beenakker's user avatar

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