For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

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2
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1answer
95 views

Prove that $\prod\limits_{2 < p \leq y}\left(1-\frac{2}{p}\right)\sim\frac{D}{\log ^2 y}$ [duplicate]

I'm writing my bachelor thesis about Brun's sieve method and his theorem. In one proof I found this statement without further explanation. It is important to show that the product doesn't converge "...
0
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0answers
24 views

Product of directed partial orders

Is a product poset (with componentwise order) of nonempty posets a dcpo if and only if each multiplier is a dcpo? (for both binary and arbitrary products)
0
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0answers
26 views

Is there a constant $C$ such that $\prod_{p\leq x}\dfrac{p-1}{p}<\dfrac{e^{-\gamma}}{\log x}\cdot C$?

By Mertens' third theorem: $$\prod_{p\leq x}\dfrac{p-1}{p}\sim\dfrac{e^{-\gamma}}{\log x}$$ But does there exist a constant $C$ such that: $$\prod_{p\leq x}\dfrac{p-1}{p}<\dfrac{e^{-\gamma}}{\log x}...
0
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0answers
31 views

I am looking a comparison of this computation and Riemann's approach for $lcm(1,2\ldots,x)$

Looking a comparison with a reasoning due to Riemann, I ask to me about the behaviour as $x\to\infty$ of the following arithmetical function $$ \left( \prod_{n\leq x}n^{-\mu(n)}\right)\cdot \left( \...
2
votes
1answer
51 views

Why does the product of adjugates equal an adjugate of the product?

How can I show that $\mathrm{adj} (AB) = \mathrm{adj}(B)\ \mathrm{adj}(A)$? It is obvious if determinants are non-zero, but if any of the matrices are singular, I just don't get it. UPD. I've just ...
0
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0answers
23 views

Prove Euler characteristic satisfies $\chi(X \times Y)=\chi(X)\chi(Y)$ for polyhedra $X$ and $Y$

Prove that for any topological polyhedra, $X$, $Y$, the product $X \times Y$ has the Euler characteristic $\chi(X \times Y)=\chi(X)\chi(Y)$ I know that for polyhedron $P$ which is homemorphic to a ...
0
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0answers
30 views

Formula for combinations involving product notation?

So after looking at the factorial formula and learning about product notation, I recognized this relation between them: $$\prod_{n=1}^kn=k!$$ And after fooling around and doing some trial and error, I ...
1
vote
1answer
38 views

“Binary-Like” Function?; In Consecutive Products as Multi-Factorials…

Summary Is there a function $Z(a,b)$ or how would one find such a function so that for $a,b\in \mathbb N$, it would produce $0$'s on for each $a$th step for each $b$th value? For example: $a=2$, ...
9
votes
1answer
219 views

How many numbers $ N \le 10^{10}$ are the product of $3$ distinct primes?

How many numbers $ N \le10^{10}$ are the product of $3$ distinct primes? I can realistically calculate any $\pi(n), n < 10^{15} $ but I don't think it's possible to list all primes $>10^8$ in ...
2
votes
2answers
32 views

Gamma representation of certain sequence

I'm trying to find a gamma rep for $ 15 \cdot 13 \cdot 11 \cdot 9 \cdot 7 \cdot ... $ Steps so far: It's a simple sequence of $ n \cdot (n-2) \cdot (n-4) \cdot (n-6) \cdot (n-8)... $ and so on. ...
1
vote
0answers
15 views

Where can I find methods to evaluate products?

I found it was slightly difficult to find resources that discussed methods for evaluating products, like $\Pi_{n=0}^ka_n$ Preferably, I want to start with the basics and move through some readings on ...
1
vote
1answer
37 views

Complex inner product proof

I have just solved this problem in the real inner product space with $\langle \cdot , \cdot \rangle$ as the inner product. Show that in a real inner product space we have: $\langle x,y \rangle = \...
0
votes
1answer
32 views

Nice formula for a sum product

So suppose I have an ordered set of numbers: $(a_1, a_2, ..., a_n)$ and I want to express the following sum/product in an elegant manner: $ a_1 + a_1 a_2 + a_1 a_2 a_3 + ... + a_1 a_2 ... a_n $ I ...
0
votes
0answers
26 views

Resolving Zeros in Product of items in list.

Given the formula: $\sqrt [ 1/N ]{ \prod _{ n=1 }^{ N }{ { P }_{ n } } } $ where ${ P }_{ n }$ is a list of real numbers, e.g. [0.4, 0.3, 0.2, 0.1] And the ...
1
vote
1answer
45 views

Finding an upperbound for $\sum_{i=2}^{n}\bigg(\prod_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$

I was wondering whether there exists a known upperbound for: $$f(n)=\sum_{i=2}^{n}\bigg(\prod_{k=2}^{i}\dfrac{p_k-2}{p_k}\bigg)$$ For example: $$f(4)=\dfrac{1}{3}+\dfrac{1\cdot3}{3\cdot5}+\dfrac{1\...
1
vote
1answer
95 views

How many groups of order $2016$ exists, which are a direct product of smaller groups?

There are $6538$ groups of order $2016$ upto isomorphism. How many groups of order $2016$ are a direct product of (at least two) smaller groups ? I calculated an upper bound by summing the ...
1
vote
1answer
42 views

Closed formula for finite product series

I need to solve the recurrence: $$ \begin{align*} T(n) &= kT\left(\frac{n}{2}\right) + (k - 2)n^3 \\ &\textit{where}\; k \in \mathbb{Z}: k \geq 2 \\ &= k(kT\left(\frac{n}{...
5
votes
1answer
58 views

Convergence of $\prod (1+ta_n)$ implies convergence of $\sum a_n$ and $\sum a_n^2$

Let $a_n$ be a sequence of real numbers and assume that $\prod _n(1+ta_n)$ converges for two non-zero values of $t$, say $t_1, t_2\in \mathbb R\setminus \{0, -1/a_1, \ldots, -1/a_i, \ldots \}$. ...
0
votes
1answer
29 views

Showing multiplication inequality using induction

Use induction to show that: $$\frac{1}{2}\frac{3}{4}\dots \frac{2n-1}{2n} < \frac{1}{\sqrt{3n+1}} $$ for $n > 1$.
1
vote
1answer
59 views

Triples of natural numbers with same sum and product

Im looking at pairs of triples of natural numbers without repititions such that the sums of the two triples are equal and the products of the two triples are equal. To be precise: Let $x<y<z$ ...
3
votes
2answers
281 views

Is knowing the Sum and Product of k different natural numbers enough to find them?

Can we uniquely identify the set of k different natural numbers (no two are the same) by knowing only their sum and product (and the value of k itself)?
0
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1answer
32 views

Expressing a product in terms of the sum

While solving a problem, I got to the expression $$(-a+b+c)(a-b+c)(a+b-c).$$ I would like to express it in terms of the sum $a+b+c$. Is there any possibility?
2
votes
2answers
87 views

Sum of all Products on Catalan numbers

how can I simplify this? let: $$ C_n = {{2n \choose n}\over n+1} $$ find: $$ \sum_{P_1 + P_2 + ... + P_k = r} \left(\prod_{j = 1}^k C_{P_j}\right) $$ thanks!
2
votes
1answer
34 views

How can I express such a product?

I know for example that $$\prod^{k}_{n=0} a_n = a_0 \cdot a_1 \cdot a_2 \cdot a_3 \cdots a_k$$ But what if I wanted to express $\space 3^k$ as a product? I know it sounds like a simple question, ...
3
votes
1answer
42 views

What is condition that the sum of $n$ complex numbers eaquals their product

Let $n\geq2$ and let $\{z_1,\dots,z_n\}$ be a set of complex numbers. Is there a condition on the $z_i$'s such that $$\sum_{i=1}^n z_i=\prod_{i=1}^n z_i$$ is identically true? For $n=2$ the ...
1
vote
1answer
23 views

Limit of products in $x_n = 1-An^{-\alpha}$ and their summation

Suppose that we have $A >0, \alpha >0$, and for each $n$, define $x_n = 1-An^{-\alpha}$ such that for large $n$ we have $x_n \in (0,1)$. Also, define the product sequence, $y_n = \prod_{i=0}^n ...
0
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0answers
30 views

Which kind of product do we have here?

The following GAP-output ...
6
votes
1answer
106 views

Simplify Product of sines

Is there a way simplify this product? $$ \sin\left({n} \frac{\pi}{2}\right) \sin\left({n} \frac{\pi}{3}\right) \sin\left({n} \frac{\pi}{4}\right) ...\sin\left({n} \frac{\pi}{n-1}\right) $$ And, is ...
0
votes
1answer
20 views

Elementary proof: division by integer makes real number smaller.

It's been so long since I took discrete math, maybe someone here can help me with what ought to be a straight forward and simple proof. Effectively I want to show this: Let a and b be positive ...
3
votes
1answer
59 views

How can I prove that every finite product can be transformed to the given form?

Suppose, the permutations $a=(123)$ , $b=(12)(34)$ , $c=(12345)$ and $d=(12)(35)$ are given. I checked with GAP that the elemts $$a^jb^kc^ld^m$$ with $0\le j\le 2$ , $0\le k\le 1$ , $0\le l\le 4$ , $...
1
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2answers
46 views

cup product in relative cohomology; why the subsets $A$ and $B$ of $X$ have to be open?

I have a question about cup products in relative cohomology. In lecture we defined the cup product on singular cohomology as follows: Let $R$ be a commutative ring with unit $1_R$, let $X$ be a ...
5
votes
0answers
35 views

Prove $\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kx)\,dx$

Let $N>0$ and $a_0,a_1,...,a_N$ be any positive numbers. How to prove that $$\dfrac{1}{2}+\sum_{n = 1}^{\infty}\prod_{k = 0}^{N}\text{sinc}(a_kn) = \int_{0}^{\infty}\prod_{k = 0}^{N}\text{sinc}(...
0
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0answers
25 views

Approximate product by product

Let $\mathbb A _n = \{a_1, \ldots, a_n\} \subset \mathbb R_+$. For given $n, K \in \mathbb N$ can we bound from above the following: $$\left|\prod _{k=1}^K x_k - \prod _{\ell=1}^Lb_\ell \right| \leq f(...
0
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2answers
82 views

Pseudocode of brute-force algorithm that finds largest product of two numbers in a list [closed]

This one will require a basic knowledge of some computer science concepts. I am trying to come up with a pseudocode brute-force algorithm that finds the largest product of two numbers in a list $a_1, ...
2
votes
2answers
79 views

Linear approximation to the product: $\prod_{k=0}^r\left(1+\frac12\left(\frac{\frac12+k+1}{\frac12+k}-\frac{\frac12+k}{\frac12+k+1}\right)\right)$

I have come upon with the next expression: \begin{equation} P_r=\prod_{k=0}^r \left(1+\frac{1}{2}\left(\frac{\frac{1}{2}+k+1}{\frac{1}{2}+k} -\frac{\frac{1}{2}+k}{\frac{1}{2}+k+1}\right)\right) \end{...
0
votes
1answer
30 views

Why the usage of $H$ in the dot product $x^H y$ instead of $T$ or $'$ for transpose?

Given two vectors $v, u \in \mathbb{R}^n$ (i.e. column vectors), then the dot product of them is defined like this $$\sum_{i=1}^n v_i * u_i$$ Or usually it can be expressed in matrix-product ...
2
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0answers
42 views

How can I find the elements generating a group in a special way?

Suppose, a finite permutation group G is given. I want to find the minimal set $x_1,...,x_n$ such that every element of $G$ can be uniquely written in the form $$x_1^{j_1}...x_n^{j_n}$$ with $0\le j_i\...
1
vote
3answers
39 views

Prove that $\prod_{n \in \mathbb{N}}{(1-a_n)} \geq 1 - \sum_{n \in \mathbb{N}}{a_n}$

The following proof is obtained from this paper. My question is how to obtain the inequality. My guess is because of the following inequality: $$\prod_{n \in \mathbb{N}}{(1-a_n)} \geq 1 - \sum_{...
0
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0answers
11 views

To clear for variable 'a' in a sum of dependent products

I can't seem to find a way to clear this equation for variable $a$: $E[k] = \displaystyle\sum_{k=1}^nk\frac{a}{n+a-k}\displaystyle\prod_{i=0}^{k-1}1-\frac{a}{n+a-i}$ Do you think it's possible? Any ...
2
votes
0answers
45 views

Is $(\omega \times \omega)^{\omega}\cong \omega \times \omega \times… \cong \omega^{\omega}$? Where “$\cong$” means homeomorphic.

I'm interested in the circumstances for when we can conclude that two ordinal spaces are homeomorphic by an examination of their written form. Specifically, I'm taking an ordinal space, say $(\omega^...
0
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0answers
35 views

speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$

I would like to find the speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$ for any $z$. For example, if $|z| < 1$, doing taylor expansion we know that it is roughly $e^{zn}.$ But I need it ...
0
votes
3answers
47 views

universal property of product: must any map satisfying it be a morphism

I am thinking about the universal property of products: Let $X$ and $Y$ be objects of a category $D$. The product of $X$ and $Y$ is an object $X \times Y$ together with two morphisms $\pi_1 : X \...
3
votes
2answers
90 views

Product of Primes

Let $\mathbb{P}$ denote the set of prime numbers. How would one evaluate $$\prod_{p\in \mathbb{P}}\frac{p-1}{p}$$ I do not think that the fact that $$\prod_{n=2}^{\infty}\frac{n-1}{n}=\lim_{n\to\infty}...
2
votes
1answer
67 views

Evaluation of $\prod^{n}_{r=1}\sin \left(\frac{(2r-1)\pi}{2n}\right)$

Find value of $$\prod^{n}_{r=1}\sin \left(\frac{\left(2r-1\right)\pi}{2n}\right)$$ Where $n\in \mathbb{N}$ and $n>1$ $\bf{My\; Try::}$ Let $$P = \sin \left(\frac{\pi}{2n}\right)\cdot \sin \left(\...
0
votes
1answer
30 views

What is the name for this product?

I have a vectors like: $\vec{a} = [a_1, a_2] $ $\vec{b} = [b_1, b_2] $ And I need a vector of products of unique combinations like: $\vec{p} = [a_1 b_1, a_1 b_2, a_2 b_1, a_2 b_2]$ does exist a ...
1
vote
1answer
59 views

Product of two sums, one finite and one infinite

I'm working on a problem and I'm not sure how to find the product of these two sums: $\left(\sum_{k=0}^{\infty}\text{something}\right)\left(\sum_{k=n}^{n}\text{something else}\right)$ The "something"...
1
vote
1answer
98 views

What symbol is used for product topology?

Let $((X_k,\tau_k))_{k \in N}$ be topological spaces. The product topology $\tau$ on $X = \prod_{k \in N} X_k$ is the coarsest topology that makes all projections $\pi_k:X \to X_k$ continuous. Is ...
0
votes
0answers
31 views

Number of optimas of product of convex functions

I am dealing with a function, which is a product of two strongly convex functions, and trying to determine the number of its local minimum. For example, I have $$H=f(x)\cdot g(x)$$, in which both $f$ ...
3
votes
1answer
41 views

Simple formula for the $n$-ary version of $(x,y) \mapsto \frac{x+y}{1-xy}$

Let $x * y = \frac{x + y}{1 - xy}$. I want a single formula for $x_1 * x_2 * \ldots * x_n$, for all natural $n$. In order to generate plausible candidates, let's see what happens at small values of $...
0
votes
0answers
77 views

Countable product of first/second countable spaces is first/second countable.

Let $N$ be a countable indexing set, and $((X_k,\tau_k))_{k \in N}$ topological spaces. Define $X = \prod_{k \in N} X_k = X^N$ and let $\tau$ be the product topology on $X$ induced by $\tau_k$s. ...