55
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 ...
- 158k
24
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 '(...
- 1,475
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 ...
- 158k
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 ...
- 3,375
16
votes
Accepted
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 ...
- 83k
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
$$
\sum_{n=1}^N \sin nt = 0 , \quad\quad\quad\...
- 17.6k
16
votes
A new way of approaching the pole of the Riemann zeta function - and a new conjectured formula
The first formula is trivial. $$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)^...
- 3,088
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 ...
- 968
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 $...
- 94.8k
14
votes
Accepted
How to prove this identity on double summation series?
Both are equal to $(\pi^2-6\log^22)/12$.
The inner sum on the right
$$\sum_{n=0}^\infty\frac{(-1)^m}{(n+1)(n+m+2)}=\frac{(-1)^m}{m+1}\sum_{n=0}^\infty
\Bigl(\frac{1}{n+1}-\frac{1}{n+m+2}\Bigr)=
\frac{...
- 6,803
14
votes
Accepted
Series involving factorials
The sum
$$\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+...
- 14.5k
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
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
$$f(x)=\sum_{n=0}^{\infty} (-1)^n (2n+1) e^{-\pi x (n+\frac12)^2}.$$
...
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,...
- 91.9k
13
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 ...
- 91.9k
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 $h(29)=h(37)=h(41)=1$.
...
- 19.9k
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
$$0 < \zeta(s) -...
- 7,262
12
votes
Accepted
Asymptotics of a special function
Gradshteyn & Ryzhik equation 3.773.1 gives (for $q>0$)
$$\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}\...
- 158k
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 ...
- 10.7k
12
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 ...
- 91.9k
12
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 $_{1}F_{1}$)...
- 1,475
12
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 ...
- 158k
11
votes
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
11
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 ...
11
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 ...
- 10.3k
11
votes
Accepted
A tantalizing Gamma quotient to challenge the Rohrlich-Lang Conjecture
It turns out that triplication is not needed here:
the recursion $\Gamma(z+1) = z \Gamma(z)$, the reflection formula
$$
\Gamma(z) \Gamma(1-z) = \frac\pi{\sin \pi z},
$$
and the duplication formula
$$
...
- 73.8k
11
votes
Сlosed formula for $(g\partial)^n$
In OEIS A124796 I considered a similar problem of computing the coefficients of $(\partial_z\circ M_g)^n$, where $M_g$ is the operator of multiplying by $g(z)$.
It turns out that the coefficients ...
- 27.6k
11
votes
Do infinitely nested radicals have any applications?
The first application goes back to Archimedes. Let me explain how.
$$
\underbrace{\sqrt{2+\sqrt{2+\ldots+\sqrt{2}}}}_{n}\to 2.
$$
The question is how fast. It turns out that the rate of convergence ...
- 24.4k
11
votes
Accepted
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+...
- 158k
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