About $\zeta(2n+1)$ There’s lots of successful methods for the finding of $$\zeta(2n)=\sum_{m\ge 1}{\frac{1}{m^{2n}}}$$ e.g Fourier series. My question is why all those methods fail for  $$\zeta(2n+1)=\sum_{m\ge 1}{\frac{1}{m^{2n+1}}}?$$ Is there a link between $\zeta(2n+1)$ and $\zeta(3)$ for all non zero $n$?
 A: Your question is "why the usual analytic methods fail for $\zeta(2n+1)$" ? The answer lies very deep, and actually it reveals a hidden algebraic aspect of $\zeta$, which is all the more astonishing as $\zeta$ is a priori an analytic object. Let me try to explain, starting from the completed $\zeta$-fuction defined by $\Lambda(s):=\zeta_{\infty} (s)\zeta(s)$, where $\zeta_{\infty} (s):= \pi^{-s/2} \Gamma(\frac s2)$. Then $\Lambda$ satisfies the functional equation $\Lambda(s)=\Lambda(1-s)$. An integer $m$ is called critical for $\zeta$ if both $\zeta_{\infty} (s)$ and $\zeta_{\infty} (1-s)$ have no pole at $s=m$. The set of critical integers for $\zeta(s)$ is {$..., 1-2n,..., -3,-1$} $\cup$ {$2,4,..., 2n,...$}. 
1) The critical values of $\zeta(s)$ are classically known (see e.g. Neukirch, ANT, Springer 322 (1999), VII,$1$) : $\zeta(1-2n)=-B_{2n}/2n$ and $\zeta(2n)=(-1)^{n+1}(2\pi)^{2n}B_{2n}/2(2n)!$ , where $B_{2n}$ are the even Bernoulli numbers, which  appear as rational coefficients in the power series expansion of $z/e^z -1$. A first stunning feature is that their numerators and denominators happen to have an interpretation in terms of Quillen's $K$-groups $K_*(\mathbf Z)$. Recall that $K_*$ is a functor from the category of commutative rings to the category of abelian groups, but its general definition actually comes from algebraic topology! Anyway, for all  $n\ge 1, \mid \zeta(1-2n)\mid = \mid K_{4n-2}(\mathbf Z)\mid/\mid K_{4n-1}(\mathbf Z)\mid$ up to powers of $2$ (an observation of Lichtenbaum, 1973).
2) The non critical values are truly the mysterious ones. It is worth noting that the odd Bernoulli numbers $B_{2n+1}$ are null for $n\ge 1$, so there cannot be any "naive" extension of the critical formulas to $s=2n+1$. For any integer $n\ge 1, \zeta(s)$ has simple pole at $s=-2n$, so what matters is actually the leading term  $\zeta^*(-2n):= lim (s+2n)\zeta (s)$ when $s$ tends to $-2n$, and we can concentrate on $\zeta^*(-2n)$ instead of $\zeta (2n+1)$. Following his previous observation, Lichtenbaum conjectured that $\zeta^*(-2n)=\pm R_{2n-1}(\mathbf Q)\mid K_{4n-2}(\mathbf Z)\mid/\mid K_{4n-1}(\mathbf Z)_{tors}\mid$ up to powers of $2$, where $R_{2n-1}(\mathbf Q)$ is the so called Borel regulator, analogous to the Dirichlet-Dedekind regulator in the analytic class number formula, but coming from differential geometry! This conjecture is now a theorem of Bloch-Kato (1990).  
3) Note that Lichtenbaum originally proposed his conjecture for all number fields, not only for $\mathbf Q$, and this has been also proved for abelian number fields (1996). It is impossible to give here even an idea of the proof because, as usually happens when a problem lies too deep, the solution is to embed it into an even deeper, but more general problem, in order to be able to mobilize machinery coming from other domains. This is what happened for the special values of $\zeta$, the "catch all" conjecture being the so called Tamagawa number conjecture of Bloch-Kato for motivic L-functions. An introduction can be found in the proceedings of the 2012 Pune conference on "The Bloch-Kato conjecture for the Riemann Zera Function", London Math. Soc., 418 (2015).
A: "All those methods, e.g. Fourier series" can evaluate
$$
\sum_{k \ne 0} \frac{1}{k^s}
$$
in closed form for $s=1,2,3,4,\dots$.  In case $s$ is even, we may divide by $2$ to get an interesting answer for $\sum_{k=1}^\infty$ involving a power of $\pi$.  In case $s$ is odd, we get zero.
A: Look at the derivatives at $0$ of
$$\sum_{n\ne 0} \frac{1}{(z+n)^2} = \frac{\pi^2}{\sin^2 \pi z}-\frac1{z^2}$$
$$\sum_{n\ge 1} \frac{1}{(z+n)^2} = \frac{\Gamma''(z+1)\Gamma(z+1)-\Gamma'(z+1)^2}{\Gamma(z+1)^2}$$
The former is an elementary function, the latter is a special function.
