# Tag Info

### 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 ...
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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 '(...
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### 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 ...
• 158k
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### 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 ...
• 3,375
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### Why does $\sum_{p=1}^n \exp\left(\frac{i\pi p l}{2m}\right)/\prod_{k=1,k\neq p}^n\sin\left(\frac{\pi (k-p)}{2m}\right)$ vanish?

Let us consider the case when $n$ is odd. Let $$P(x):=\exp\left(\frac{ixl}{2}\right)\left(\exp\left(\frac{ix}{2}\right)-\exp\left(\frac{-ix}{2}\right)\right),$$ and notice that the degree of $P$ as an ...
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### 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 ...
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### 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 ...
Accepted

• 91.9k
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### 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 ...
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### 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 $h(29)=h(37)=h(41)=1$. ...
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### 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 ...
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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 ...
• 91.9k
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### 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 $_{1}F_{1}$)...
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### 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 ...
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### The function $\sum_{0}^{\infty} x^n/n^n$

There is a paper of G. H. Hardy, where this function is studied in great detail: G. H. Hardy, On the integral function $\Phi_{ a,\alpha,\beta}(z)=\sum x^n/(n+a)^{\alpha n+\beta}$, Quarterly J. Math.,...
• 83.1k
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### 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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### 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 ...
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### Erroneous Wolfram result for $\sum_{k=1}^\infty (k^3 + a^3)^{-1}$, looking for correct formula
I think the statement in the OP that $W_2(a)$ and $W_3(a)$ remain bounded when $a\rightarrow 0$ is mistaken, so that there is no inconsistency with the Mathematica result. The three roots of \$(w+1)^3+...