What is known about the pattern for $\zeta(2n+1)$?

sequences-and-series riemann-zeta

Solution 1:

As Qiaochu writes in his comment, $\zeta(2n+1)$ (for $n \geq 1$) is expected to be algebraically independent of $\pi$, and furthermore for different values of $n$ these numbers are also expected to be algebraically independent of one another.

These expectations are far from being proved (as far as I know), but are not idle speculations: contemporary number theorists have a very tightly woven web of conjectures about values of $\zeta$-functions, and $L$-functions, which is supported by large amounts of theoretical and experimental evidence, and I don't think there is any reason to doubt that the expectations are correct.

Given this, I don't expect that the numbers $f(2n+1)$ will follow any significant pattern.

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