$\zeta(z)$ Identities and a Conjecture

Question

Brief Introduction

I found something quite interesting and would like to share it here.

The surface area and volume of a unit radius $n$-sphere can be expressed in terms of the gamma function $\Gamma(z)$ as follows. Let's define the surface area $$S(n) = \frac{2\sqrt{\pi^n}}{\Gamma\left(\frac{n}{2}\right)}$$ and the volume $$V(n) = \frac{\sqrt{\pi^n}}{\Gamma\left(\frac{n}{2}+1\right)}$$ Now let's define a function $\theta_k(n)$ where $k$ is a 'dimensional shift' such that $S(n-k)$.

$\theta_k(n)$ can be expressed as a ratio as follows $$\theta_k(n) = \frac{S(n-k)}{V(n)}$$ $$ = \frac{n\Gamma(\frac{n}{2})}{\sqrt{\pi^k}\Gamma\left(\frac{n-k}{2}\right)}$$

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There are many potentially interesting properties that $\theta_k(n)$ exhibits. Many identities like $$\theta_0(n)=n$$ $$\theta_{2k}(n)\theta_{-2k}(n) = n(n-2k)\prod_{m=1}^{k-1}\frac{n-2m}{n+2m}$$ $$\theta_{-k}(n)= \pi^k \theta_{k}(n)$$ There is one that we will use for the next few parts and has many interesting relations, namely $$\theta_{2n}(2n-1) = -\frac{\Gamma(n+\frac{1}{2})}{\pi^n \sqrt{\pi}}$$ $\theta_k(n)$ in general appears in many different areas of mathematics

An example: $$\sqrt{\pi}\int_{-\infty}^\infty x^{2n+2}e^{-x^2}\,dx=-\frac{\theta_{2n}(2n-1)}{2\left(-\pi\right)^n}$$

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Identities With The Riemann-Zeta Function

1. $$\frac{\zeta (\frac{1}{2}-n)}{\zeta(n+\frac{1}{2})} = -i^{n(n+1)}\frac{\theta_{2n}(2n-1)}{2^n}$$

2. $$\theta_{2n}(2n-1) = -\frac{(2n-1)!!}{(2\pi)^n}$$ which therefore implies $$\frac{\zeta (\frac{1}{2}-n)}{\zeta(n+\frac{1}{2})} = i^{n(n+1)}\frac{(2n-1)!!}{(4\pi)^n}$$

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Next I will present a conjecture based off of a single observation regarding the ratio of $\theta_k(n)$ and $\zeta(n)$.

Conjecture

Let's start off with the following equation $$\frac{2^{n}}{\theta_{2n}(2n-1)} = \phi(n) \zeta(n)$$

We are interested in finding the values of $\phi(n)$, namely for $\text{parity}(n) := \text{even}$ as these are rational.

(I managed to derive $\phi(2n) = (-1)^n\frac{2(4n)!!}{B_{2n}(4n-1)!!})$ however am unsure of whether this is correct or not for all $n \in \mathbb{Z}$

A few of the first values of $\phi(2n)$ (ignoring the $(-1)^n$ term) $$\phi(0) = 2$$ $$\phi(2) = 32$$ $$\phi(4) = \frac{1536}{7}$$ $$\phi(6) = \frac{4096}{11}$$ and strangely $$\theta_3(12) = \frac{1536}{7\pi^2}$$ This seems like quite a big coincidence. The conjecture states that $$\theta_k(n) = \pi^{-\alpha}\phi(\beta)$$ for some $n, k$ and $\alpha, \beta \in \mathbb{N}$

This conjecture being true would imply that any $\zeta(2n)$ can be expressed in terms of $\theta_k(n)$ (and $\pi$) alone, like demonstrated below

$$\frac{\theta_{8}(7)\theta_3(12)\pi^{2}}{2^{4}} = \frac{1}{\zeta(4)}$$

My question is simply: Is the conjecture true or not?

This is quite a big post and there are bound to be some mistakes here and there, so please correct any if you see some. The objective of this post is 1. to share a potentially interesting result, and 2. Ask whether the conjecture is true or not. If anyone has anything to contribute regarding the post, perhaps an application of $\theta_k(n)$ in a completely different field of mathematics, some identities with other well-known functions, etc then feel free to post it below. Thank you everybody for your patience

Edit: Thanks to Steven Clark for the corrections

(I have shortened the post by removing all of the lengthy proofs)

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Bounty (Expired)

Bounty for anyone who can prove (or disprove) this conjecture. Good luck!

Answer 1

If I understand correctly ,you want $\zeta(2n) = \frac{\Gamma(n)}{\Gamma((n+k)/2)} 2^a \pi^b$

for integer $a,b,k$.

$\zeta(6) = \zeta(2*3) = \frac{\pi^6}{945} = \frac{\Gamma(3/2)}{\Gamma((3+8)/2)} 2^4 \pi^6$

works where the gamma part is your " special function ".

It fails for $\zeta(2),\zeta(4)$ and many more. In fact I think $\zeta(6)$ is the only case that works.

Notice that the bernouilli numbers are complicated and very different from factorials. The bernouilli numbers have no closed form in terms of a ratio of gamma functions. Notice we also need to check only a finite amount of $k$ values for each $\zeta(2n)$ since $n$ is fixed and too high or too low $k$ could never work.

Notice that for small $n$ we have $\zeta(2n) = 1/a_n \pi^{2n}$ , so unit fractions for the first cases. This makes a ratio of gamma's very unlikely.

So your conjecture is probably false.