Apery's constant

Question

I read that it is unknown if $\zeta (3)$ is algebraic but it is known to be irrational. Has anyone proved anything of the form $\zeta (3)$ is not a root of a polynomial of degree $12345$ with integer coefficients, or is it that the degree one case proof won't work for any other degrees?

Answer 1

This post is more of a comment as opposed to an answer

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Apéry's constant was proven to be irrational in 1978 via Apéry's Theorem,

$$ζ(3) = \sum_{n = 1}^{\infty}\frac{1}{n^3} = \frac{5}{2}\sum_{n = 1}^{\infty}\frac{(-1)^{n - 1}}{n^3{{2n}\choose{n}}}.$$

But for an easy irrationality proof of Apéry's constant, the formula below is most commonly given:

$$ζ(3) = \frac{7\pi^3}{180} - \sum_{k = 1}^{\infty}\frac{2}{k^3(e^{2\pi k} - 1)} \tag*{$\because \pi \land e \neq \frac{p}{q}$}$$

with the exception of Ramanujan's formula of the Dirichlet Lambda Function $λ(z)$, created in 1985:

$$λ(3) = \sum_{k = 0}^{\infty}\frac{2}{(4k + 1)^3} - \frac{\pi^3}{64} = \frac{7}{8}ζ(3).$$

But the most common formula used to express Apéry's constant in particular is

$$ζ(3) = \left\{\frac{4}{3}\sum_{k = 0}^{\infty}\frac{(-1)^k}{(1 + k)^3} \lor \sum_{k = 1}^{\infty}\frac{H_n}{2n^2} : H_n = \sum_{k = 1}^{n}\frac{1}{k} = \text{Harmonic Number}\right\}.$$

I do not know of any formulae that expresses the value of $ζ(3)$ to be the root of a polynomial to the degree of $12345$, but there are three formulae I know which have several terms included in it being raised to various degrees:

$$ζ(3) = \sum_{n = 0}^{\infty}\frac{(-1)^nn!^{10}(205n^2 + 250n + 77)}{64(2n + 1)!^5}$$

$$= \frac{1}{4}\sum_{n = 1}^{\infty}\frac{(-1)^{n - 1}n^3{{2n}\choose{n}}{{3n}\choose{n}}(56n^2 - 32n + 5)}{(2n - 1)^2}$$

$$= \sum_{n = 0}^{\infty}\frac{(-1)^n(5265n^4 + 13878n^3 + 13761n^2 + 6120n + 1040)}{72{{3n}\choose{n}}{{4n}\choose{n}}(n + 1)(4n + 1)(4n + 3)(3n + 1)^2(3n + 2)^2}.$$

But if you want $ζ(3)$ to be expressed as a root then you can use this formula, created in 1997:

$$ζ(3) = \sqrt {2\bigg(\frac{\pi^6}{540} - \sum_{n = 1}^{\infty}\frac{H_n}{n^5}\bigg)}.$$

Answer 2

This is a COMMENT and quote I consider of interest for many people. This remarkable result was given in the ICM, 1978, Helsinski, if I remember correctly.