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.

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    $\begingroup$ 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. $\endgroup$
    – J.G.
    Nov 26 '20 at 20:37
  • $\begingroup$ @J.G. Thanks for the links! I should have checked the Wiki, didn't think I could find it there! $\endgroup$ 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.


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