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5 answers
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Closed-form expression for certain product

$\mathrm G$ is Catalan's constant. I recently found the product $$ \alpha=\prod_{n=1}^{\infty}\frac{E_n(\frac12)E_n(\frac7{12})E_n(\frac1{20})E_n(\frac{13}{20})}{E_n(\frac14)E_n(\frac1{12})E_n(\...
clathratus's user avatar
16 votes
1 answer
502 views

Conjecture on sum over permutations of products of Catalan numbers

Context In a recent paper involving entanglement in linear optics, we came across some summations involving Catalan numbers and permutations. In particular, these sums arise when doing integration ...
Joseph Iosue's user avatar
12 votes
3 answers
3k views

Do disjoint unions and fiber products commute?

Do disjoint unions and fiber products commute? In other words, is the following statement true? Statement: Let $C$ be a category with (infinite) coproducts and fiber products. Let {$U_{i}$} be a ...
Hiro's user avatar
  • 945
11 votes
2 answers
741 views

Bounding Euler products (or almost) by products of zeta functions

Let $s_1, s_2 \in (1/2,1\rbrack$. I would like to bound the product $$A=\prod_p \left(1 + \frac{p^{-s_1} p^{-s_2}}{(1-p^{-s_1}+p^{-1}) (1-p^{-s_2}+p^{-1})}\right)$$ Now, I am almost positive that $...
H A Helfgott's user avatar
  • 19.3k
10 votes
2 answers
983 views

Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?

This question is an old question from mathstackexchange. Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $ And let $ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $ It appears that we have $$\sup ...
mick's user avatar
  • 703
8 votes
2 answers
995 views

An interesting infinite product involving the factorial function with connection to the K and gamma function

I have posted this question in StackExchange, but it didn't get any answers there. This question is important for my research. I got stuck on an infinite product which even WolframAlpha can't answer. ...
user avatar
7 votes
1 answer
423 views

Factorial-based constant

Am looking for a name for: $$\prod\dfrac{1}{1-\dfrac{1}{n!}}$$ $$=2.529477472079152648180116154253954242$$ Wolfram|Alpha Expanding the formula gives: $$(1+\frac{1}{2!}+\frac{1}{2!^2}+\dots)(1+\...
JMP's user avatar
  • 1,226
7 votes
1 answer
1k views

Is a categorical coproduct of epimorphisms (monomorphisms) always an epimorphism (a monomorphism)?

Let $\mathbf{C}$ be a category (that does not necessary have a coproduct for every collection of objects). Suppose that we have two families of objects $(A_i)_{i\in I}$ and $(B_i)_{i\in I}$ in $\...
Batominovski's user avatar
7 votes
0 answers
297 views

Cartesian product is to monoidal product as pullback is to what?

I'm trying to complete the following pattern product : monoidal product : coproduct pullback : ? : pushout That is, if the monoidal product is a ...
Bruno Gavranovic's user avatar
6 votes
2 answers
2k views

Condition to ensure that the product of closed maps be closed

If $f_i : X_i \to Y_i$ with $i=1,2,\ldots,n$ are closed maps between topological space it is known that their product map $$f : X_1 \times \cdots \times X_n \to Y_1 \times \cdots \times Y_n : (x_1, \...
Richard Bonne's user avatar
6 votes
2 answers
383 views

Does the limit of this product over primes converge for all $\Re(s) > \frac12$?

Numerical evidence suggests that: $$\displaystyle F(s):= \lim_{N \to \infty}\, \ln^s\left(p_N\right)\, \prod_{n=1}^N \left(\dfrac{\left(p_n-1\right)^s}{p_n^s-1} -\frac{1}{p_n^s}\right)$$ with $p_n$ ...
Agno's user avatar
  • 4,179
6 votes
1 answer
314 views

Homology of the product of spaces with integer coefficients and the Massey products

Consider $H_*(X\wedge Y;Z)$, where $X=Y=BZ/2$ for concreteness' sake. If we write $e_i$ the generator of $H_i(BZ/2;Z/2)$., we see that the $E_2=E_{\infty}$ term of the Bockstein spectral sequence ...
user43326's user avatar
  • 3,031
6 votes
0 answers
97 views

Existence of stable spaces

An element $X$ of a class of topological spaces is called the stable space for that class if for every space $Y$ in the class we have that $X\times Y$ is homeomorphic to $X$. Note that a stable space ...
D.S. Lipham's user avatar
  • 3,055
5 votes
2 answers
991 views

When does the radius of convergence of the product of two $p$-adic power series increase?

Let $p$ be a prime number and denote by $R(f)$ the radius of convergence of a power series $f(x) \in \mathbb{C}_p[[x]]$, where $\mathbb{C}_p$ is the completion of the algebraic closure of $\mathbb{Q}...
Sandi's user avatar
  • 55
5 votes
3 answers
843 views

How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?

I tried to find the indefinite integral $$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx) \, dx$$ by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got $$ f_n(x)=-i2^{-\frac{n(n+1)}{2}-1}\int y^{-\frac{n(...
Faoler's user avatar
  • 431
5 votes
2 answers
340 views

Product of inductive limit topologies on $C_c(X)\times C_c(X)$

I have a stupid question about a topology on $C_c(X)$. Here $X$ is locally compact Hausdorff. Can assume $\sigma$-compact if it helps. Definition (topology on $C_c(X)$): For each compact $K \subset X$,...
Chertopkhanov on Malek Adel's user avatar
5 votes
1 answer
604 views

Is it possible to express the functional square root of the sine as an infinite product?

Cross-post from MSE. It is known that the sine can be expressed as an infinite product: $$\sin(x) = x \prod_{n=1}^{\infty} \Big{(} 1 - \frac{x^{2}}{n^{2}{\pi}^{2}} \Big{)} .$$ We can define that ...
Max Muller's user avatar
  • 4,485
5 votes
1 answer
275 views

Does the category of Lawvere theories have products?

I know Law has a tensor product, is closed with respect to that tensor product, and it has coproducts. Does it have products? My best guess at the cartesian product of Lawvere theories is the "...
Mike Stay's user avatar
  • 1,532
5 votes
2 answers
453 views

Function with zeros plus/minus the primes

While playing with Cohen's pari script prodeulerrat found a function. For $s \in \mathbb{C}$ define $$ f(s) = \prod_{p \text{ prime}} (1-\frac{s^2}{p^2})$$ The product converges everywhere, no poles ...
joro's user avatar
  • 24.2k
5 votes
2 answers
380 views

Intuitive explanation of regularized products

I've come across some regularized product during study of zeta regularization . We can prove various results like : $ \infty != \prod_{k=1}^\infty k = \sqrt{2\pi} $ I also know the proof using $\...
user avatar
5 votes
0 answers
167 views

Dual Steenrod squares

Fix the ground ring $\mathbb{F}_2$ and let $X$ be a space with finite homology. Then we have an isomorphism $\Phi^i_X:H_i(X)\to H^i(X)^*,a\mapsto \langle-,a\rangle$ which allows us to define the dual ...
FKranhold's user avatar
  • 1,623
4 votes
2 answers
327 views

The complex trigonometric function degenerates to the positive integer

For any integer $N \geq 2$, we have the identity: $$\frac{\ \prod _{n=1}^{N-1}\ \left(2+2\sum _{m=1}^{n\ }\cos \frac{\ m\pi \ }{N}\ \right)\ }{\prod _{n=1}^{N-1}\ \left(1+2\sum _{m=1}^{n\ }\cos \frac{\...
YanChen's user avatar
  • 43
4 votes
1 answer
254 views

Product topology from two premetric spaces induced by sum of premetrics?

For metric spaces $(M_1, d_1)$ and $(M_2, d_2)$, it is an exercise that the product topology on $M_1\times M_2$ is induced by the metric $d((x_1, y_1), (x_2, y_2)) =d_1(x_1, x_2) + d_2(y_1, y_2)$. Do ...
fsp-b's user avatar
  • 421
4 votes
1 answer
250 views

Can a commutator of a special type be conjugate to its inverse?

Let $H=H_1\ast H_2$ be a free product of non-trivial groups $H_1$ and $H_2$. We call an element $h\in H$ hyperbolic if $h\not\in H_i^g\overset{\textrm{def}}{=}\left \{ g^{-1}fg\ |\ f\in H_i \right \}$ ...
Andrey's user avatar
  • 41
4 votes
1 answer
585 views

Is the product of two supermodular functions supermodular?

The definition of Supermodularity is that for every $x′>x$ and $y′>y$, we have \begin{equation*} f(x′,y′)+f(x,y)>f(x′,y)+f(x,y′). \end{equation*} Suppose $f$ and $g$ are supermodular, non-...
JEcon's user avatar
  • 41
4 votes
2 answers
901 views

Functions that can be written as direct products of other functions; question about terminology and notation

Let $$f : X_0 \rightarrow Y_0, \;\;\; g:X_1 \rightarrow Y_1$$ and define that the "direct product" of $f$ and $g$ is a map $$f \otimes g : (X_0 \times X_1) \rightarrow (Y_0 \times Y_1), \mbox{ such ...
goblin GONE's user avatar
  • 3,693
4 votes
1 answer
160 views

How far is a countably infinite reduced abelian $p$-group from being an infinite direct sum?

Question Let $G$ be a countably infinite reduced abelian $p$-group. Is it always possible to write it has an infinite direct sums of non-trivial groups? If it is not true, how far is $G$ from being an ...
PHL's user avatar
  • 276
4 votes
2 answers
2k views

Distribution of a product of two discrete i.i.d. variables

The problem is to estimate the distribution of product of two $\textit{discretized Gaussian}$ random variables with zero means. The discretized Gaussian means that the p.m.f. looks like $D_s(x)=\...
Elena Kirshanova's user avatar
4 votes
0 answers
158 views

Which Topological Spaces are Powers?

Given a topological space $X$ and closed subspace $Y \subset X$, it may be the case that $X$ is a power of $Y$. That means $\displaystyle X = \prod_{i < \kappa} Y_i$ for some cardinal $\kappa$ ...
Daron's user avatar
  • 1,761
3 votes
2 answers
261 views

Lower bound for Euler's function

Euler function is defined, for $|x|\le 1$, as follows: $$\phi(x)=\prod_{i=1}^\infty(1-x^i)$$ Upper bounds for $\phi$ can be simply derived from ending the product early, e.g. $$\phi(x)<\prod_{i=1}^...
R B's user avatar
  • 608
3 votes
1 answer
290 views

Sum with products turned into subsequences

Let $p, q \in \mathbb{Z}$. Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and $$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$...
Notamathematician's user avatar
3 votes
1 answer
303 views

Is Spec of a ring monoidal or anti-monoidal?

Let $A$ and $B$ be rings. A very senior mathematician impressed on me the importance of writing $$ \operatorname{Spec}{A \otimes B} = \operatorname{Spec}{B} \times \operatorname{Spec}{A} $$ One can ...
David Corwin's user avatar
  • 15.1k
3 votes
1 answer
253 views

Does $\prod_{n=2}^{\infty} \left(\frac {1}{1-\frac{\chi_k(n)}{n^s}} \right)$ converge for non-principal characters for all $\Re(s) > \frac12$?

This question loosely builds on this one. Take the following infinite product: $$N(s,\chi_k)=\prod_{n=2}^{\infty} \left(\frac {1}{1-\dfrac{\chi_k(n)}{n^s}} \right)$$ with $\chi_k$ a Dirichlet ...
Agno's user avatar
  • 4,179
3 votes
0 answers
102 views

Cardinal of a set cinsist of product of two sets?

Let $$ A=\{1,2,\ldots,p-1\},\qquad B=\{1,2,\ldots,q-1\} $$ where $p,q$ are primes not necessarily distinct. Is there any elementary way to find the cardinal of the following set $$ AB=\{ab:\ a\in A,\ ...
asad's user avatar
  • 841
3 votes
0 answers
508 views

Two (strictly related) proofs by induction of inequalities

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
Vincenzo Oliva's user avatar
2 votes
2 answers
2k views

product 1+1/p in terms of Chebyshev's theta or psi function

I would like to know if there is any formula for $ \prod_{x<p\leq y}\left(1+\frac1p\right) $ in terms of $\theta$ or $\psi$ functions $ \theta(x)=\sum_{p\leq x}\log p $ and $ \psi(x)=\sum_{p^\nu\...
asd's user avatar
  • 163
2 votes
3 answers
259 views

Nonzero solutions of an infinite product

Let $-\frac{1}{2}\le a \le\frac{1}{2}$ and $b\in[0,\infty)$. Definitions: $$f_k(a;b):=\frac{(2k+\frac{1}{2}+a)^2+b}{(2k+\frac{1}{2}-a)^2+b}(\frac{k}{k+1})^{2a},$$ $$f(a;b):=\prod\limits_{k=1}^\infty ...
user90369's user avatar
  • 293
2 votes
1 answer
662 views

building a product of two categories [closed]

MacLane, as well as probably any other category book, does not hesitate to define a product of two categories as a category consisting of pairs of objects, etc. Now my question is: what law of nature ...
Vlad Patryshev's user avatar
2 votes
1 answer
424 views

Yoneda extension preserving finite products?

Let $C$ be a category and let $F:C\rightarrow D$ be a functor with $D$ locally presentable and cartesian closed. When does the Yoneda extension $\widehat{F}=Lan_{y} F:[C^{op},Set]\rightarrow D$ ...
user84563's user avatar
  • 915
2 votes
1 answer
231 views

Subsequences of odd powers

Let $p$ and $q$ be integers. Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. Then ...
Notamathematician's user avatar
2 votes
1 answer
716 views

Definition of twisted geometries and existence of coordinate transformation for twisted $AdS_2 \times S^2$

In the paper Multiply Twisted Products by Yong Wang, general definitions for so called warped and twisted products are given: A (singly) warped product $B \times_b F$ of two pseudo-Riemannian ...
horropie's user avatar
  • 649
2 votes
1 answer
153 views

Product of a vector by an inverse of Toeplitz matrix

It is well known that using fast Fourier transform it's possible to multiply a vector by a Toeplitz matrix $A \cdot v = w$ in $n\cdot\log(n)$ operations. I read somewhere that also the product of a ...
Enea Olati's user avatar
2 votes
1 answer
182 views

$\sigma$-product of the Hilbert cube

Given a homogeneous space $X$ and $p\in X$, we define the sigma product to be the following subspace of $X^\omega$: $$\sigma X=\{\mathbf x \in X^\omega:x_n=p\text{ eventually}\}$$ ("eventually&...
D.S. Lipham's user avatar
  • 3,055
2 votes
0 answers
202 views

Is there a theory of formal product series?

A few years ago, I asked a question on MSE about the existence of an infinite product representation of a functional square root of the sine function. No answers were given, though user ...
Max Muller's user avatar
  • 4,485
2 votes
0 answers
700 views

Confusing notation for sets of unordered vs ordered pairs

Given two finite sets $X$ and $Y$, one may consider the ordered pairs $(x,y)$ with $x\in X$ and $y \in Y$. Then, $(x,y) \not= (y,x)$, and $(x,x)$ exists if $x\in X$ and $x\in Y$. One may also consider ...
Matthieu Latapy's user avatar
2 votes
0 answers
1k views

Is there an infinite product like this for $\cos x$?

There are infinite products of iterated square roots for $\log x$ and $\arccos x$ as functions of $x$. For example $$\log x = \frac{x - 1}{\sqrt{x}\sqrt{\frac{1}{2} + \frac{1}{2}\left ( \frac{1 + x}{...
John Finkelstein's user avatar
2 votes
0 answers
112 views

How to estimate $\prod_{t=1}^{N}\frac{1}{2-z^t}$ for large $N$?

Based on the top answer to How to estimate of $\prod_{k=a}^N \frac{1}{e^{k\kappa}-1}$ for large $N$? Can anyone find an approximate closed form for $$ \frac{\mathrm{d}^k}{\mathrm{d}z^k}\prod_{t=1}^{N}...
apg's user avatar
  • 612
2 votes
0 answers
155 views

Two products over primes

For $k \in \mathbb{N}$ define $$ f(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k+1)}\right)$$ $$ g(k) = \prod_{\text{p prime}}\left(1+\frac{1}{p^k(p^k-1)}\right)$$ By the product for zeta $f(1)...
joro's user avatar
  • 24.2k
2 votes
0 answers
456 views

Morphisms of Spectral Sequences and alternating products

Let $E_{a,b}^{r}, F_{a,b}^{r}$ be two (co)homologica first quadrant spectral sequences of vector spaces over a field $K$, and $f : E \to F$ be a morphism of spectral sequences. Assume that morphisms $...
1 vote
2 answers
311 views

Closed form of $\prod_{i=0}^{N}\big(i!\big)^{{N}\choose{i}}$

I have made a question here about closed form of the following: $$\prod_{k=0}^{N}\big(k!\big)^{{N}\choose{k}}$$ I know that there is a known closed form for, $$\prod_{i=0}^{N}\big(i!\big)=G(N+2)$$ ...
Wiliam's user avatar
  • 155