# Specific value of $\zeta(3/2)$?

Is anything known about the value of $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}?$$ It is a classical result that $$\displaystyle \zeta(2)= \frac{\pi^2}{6}$$ and $$\zeta(3)$$ has been shown to be irrational by Roger Apéry in 1979.

Do we even know if $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}$$ is an irrational number or not? Is it true that $$\zeta(3/2)=\sum_{n\geq 1}\frac{1}{n^{3/2}}=\frac{a}{b}\sum_{n\geq 1}\frac{(-1)^{k-1}}{n^{3/2}\binom{2n}{n}}?$$ Not sure if Apéry's proof can be adapted here since I haven't read it.

• It's been studied, but not enough to be here. This list includes it, mentioning its roles in physics (one is quantified here) and linking to OEIS. I think your specific mathematical queries about it may make this question better suited to Math Overflow.
– J.G.
Nov 26 '20 at 20:37
• @J.G. Thanks for the links! I should have checked the Wiki, didn't think I could find it there! Nov 26 '20 at 20:41

By no means an answer but an extended comment. I should also preface I am an amateur mathematician with no formal education. Following Wikipedia's entry for the Riemann zeta function we have the following functional equation: $$\zeta(s)=2^{s}\pi^{s-1}\sin\left(\pi s/2\right)\Gamma(1-s)\zeta(1-s)\label{a}\tag{1}$$ Suppose $$s=3/2.$$ Via direct substitutions I can rewrite LHS side of $$\ref{a}$$ explicitly as $$2^{3/2}\pi^{3/2-1}\sin\left(3\pi/4\right)\Gamma(1-3/2)\zeta(1-3/2);\label{b}\tag{2}$$ which after regrouping and gathering like terms is equal to $$2^{3/2+1/2}\pi^{3/2-1+1/2}\zeta(-1/2);\label{c}\tag{3}$$ which is reducible to $$4\pi\zeta(-1/2).\label{d}\tag{4}$$ And so $$\zeta(3/2)=4\pi\zeta(-1/2), \label{e}\tag{5}$$ Observe:
Case 1: Note $$4\pi$$ is an irrational number. If $$\zeta(-1/2)$$ is rational then RHS side of $$\ref{e}$$ is an irrational number and so to by equality is $$\zeta(3/2).$$
Case 2: Again note $$4\pi$$ is an irrational number. If $$\zeta(3/2)$$ is a rational number then the RHS of $$\ref{e}$$ is a rational number. If $$\zeta(-1/2)$$ is an rational number then the RHS of $$\ref{e}$$ is irrational; which is a contradiction. Subsequently $$\zeta(-1/2)$$ is an irrational number.
From cases 1 and 2 it seems at least one of $$\zeta(3/2)$$ or $$\zeta(-1/2)$$ is an irrational number. In particular they cannot both be rational. More strongly it would seem of the infinite number of pairs $$\{\zeta(j/2),\zeta(1-j/2)\}_{j\in\mathbb{N}}$$ at least one number from each of the infinite pairs is irrational.