# Tagged Questions

For questions on infinite products: convergence, computation, etc...

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### On $\sum_{\substack{\zeta(\frac{1}{2}+i\gamma)=0\\0<\gamma<T}}\prod_{n=1}^\infty \left| 1-\frac{(\gamma\log x)^2}{n^2\pi^2}\right|$ as $O(\log x)$

On assumption that the identity (2) for a representation of $\pi(x)$ holds, see here Two Representations of the Prime Counting Function in this site Mathematics Stack Exchange, and since using the ...
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### What are practical applications of infinite products?

My analysis book covers a section on infinite products. So I started wondering what the practical applications of infinite products are in science and engineering, but couldn't find anything yet. Also,...
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### Principal argument summation

Let $\text{Arg}$ be a an principal argument in $(-\pi, \pi]$. I know that, for all $z_1,z_2\in\mathbb{C}\setminus \{0\}$, the expression $\text{Arg}(z_1z_2)= \text{Arg} z_1 + \text{Arg} z_2$ doesn't ...
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### Computing an infinite product $\prod_{n=1}^{\infty} \frac{1}{2}(1+\cos\frac{x}{2^n})$

I would like to compute the infinite product $\displaystyle f(x)=\prod_{n=1}^{N\rightarrow\infty} \frac{1}{2}\left({1+\cos\frac{x}{2^n}}\right)$ for a given real $x$. Since the terms in the product ...
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### Uniformly convergence on a set implies a smaller set

Here is a product, that looks like this $\prod_{n=1}^{\infty}p_n(z)$. There are two questions: If you have shown that the product is uniformly convergent on a compact subset of $\mathbb{C}$, is ...
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### How to express a homogenous function using an infinitely recursive matrix operation?

As is well known, if the multivariate function $f(\mathbf{x})$ is homogenous of degree $h$, then the partial derivatives of $f$ are homogenous of degree $h-1$. Also, say that we know $f$ is ...
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### Does this infinite product converge? And how to express it neatly?

First of all, does this product have a "nicer" functional form--i.e., analogous to how you can write geometric sums in a nice closed expression: $$(x-0)(x-1)(x-2)...(x-n)$$ Secondly, does this ...
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### What's about $\sum_{n=1}^{\infty} \frac{e^{H_n}\log H_n}{n^3}$, where $H_n$ is the nth harmonic number?

I would like to do a toy verification of the Riemann hypothesis exploiting theLagarias theorem (see the section Applications in the following link) and the fact that we know a lot of decimals for ...
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### Showing that $\sigma=\prod_{n=1}^{\infty}(n!)^{\frac{1}{2^{n+1}}}$

Somos's quadratic recurrence constant The Somos's Quadratic recurrence constant is defined by the sequence $g_n=ng_{n-1}$ with initial value of $g_0= 1$ The value of $\sigma=1.661687...$ An ...
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### $\cos(\pi x) = \prod_{n=0}^{\infty} \left( 1-\frac{x^2}{(n+\frac{1}{2})^2}\right)$?

I need to show that $$\cos(\pi x) = \prod_{n=0}^{\infty} \left( 1-\frac{x^2}{(n+\frac{1}{2})^2}\right)$$. For $x \notin \mathbb Z$ one can use $\cos(\pi x ) = \frac{\sin(2 \pi x )}{2 \sin(\pi x )}$ ...
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### Infinite product include summation

I would like to find an infinite product of$$\prod _{n=2}^{\infty} \left(1+\frac{(-1)^{n-1}}{a_n}\right)$$ where $a_n = \sum_{k=1}^{n-1} \frac{n!(-1)^{k-1}}{k!}$ I tried to compute $a_2 , a_3 ,...$,...
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### Infinite Sets Proof [closed]

I have a few questions regarding this problem below: Prove that if A and B are finite sets, then A ≈ B if and only if |A| = |B|. Would I assume that |A| = |B|? Which would obviously make A ≈ B ...
Let $S=\{n\in \mathbb{Z}_+ \mid n \equiv 1, 5 \,\,(\text{mod 6})\}.$ Let $a(n)$ be the number of partitions of $n$ into parts belonging to $S,$ and $b(n)$ be the number of partitions of $n$ into ...