Let $$I\_{N} = \int\_{n - \frac{1}{2}}^{n + \frac{1}{2}} \cos^{2N}(2\pi x)dx = \frac{2N-1}{2N} \int\_{n - \frac{1}{2}}^{n + \frac{1}{2}} \cos^{2N-2}(2\pi x)dx $$ therefore (I think)
$$ I\_N = \frac{(2N-1)!!}{(2N)!!}$$ hence $$ \frac{(2N)!!}{(2N-1)!!} I\_N = 1$$
Now let for $n \in \mathbb{N}$
 $$J\_{3,N}(n) = \frac{(2N)!!}{(2N-1)!!} \int\_{n-\frac{1}{2}}^{n + \frac{1}{2}} \frac{1}{x^3} \cos^{2N}(2\pi x) dx $$ I think (but I was unable to prove so far) that $$\lim\_{N\to \infty}J\_{3,N}(n) = \frac{1}{n^3}$$ for all $n \in \mathbb{N}$


If this is the case then 
$$ \lim\_{N \to \infty} \int\_{\frac{1}{2}}^{\infty} \frac{1}{x^3} \cos^{2N}(2\pi x) dx = \sum\_{n=1}^{\infty} \frac{1}{n^3} = \zeta(3)$$ 
Is this true? How can I provide a proof for this? Of course this can be generalised to other arguments of $\zeta$ 



On Riemann zeta function and Dirac delta function/distribution
Let $$C\_n = \int\_{-1}^1 (1-\frac{x^2}{2})^n dx \sim \int\_{-\pi/2}^{\pi/2} |\cos(x)|^n dx$$
If $f$ is continuous
 $$\lim\_{n \to \infty} \int\_{-1}^{1} \frac{(1-\frac{x^2}{2})^{n} }{C\_n} f(x)=\lim\_{n \to \infty} \int\_{-\pi/2}^{\pi/2} \frac{|\cos(x)|^{n} }{C\_n} f(x) = f(0)$$
(So that $\frac{|\cos(x)|^n }{C\_n}1\_{|x| < \pi/2} \to \delta(x)$ in the sense of distributions)


And hence if $|f|$ is integrable, decreasing and $\sum\_{n=1}^\infty |f(n\pi)| < \infty$, for $\epsilon > 0$
 $$\lim\_{n \to \infty}\int\_{\epsilon}^\infty \frac{|\cos(x)|^n}{C\_n} f(x)dx= \sum\_{n=1}^\infty f(\pi n )$$
ie. for $\Re(s) > 1$
 $$\lim\_{n \to \infty}\int\_{\epsilon}^\infty \frac{|\cos(x)|^n}{C\_n} x^{-s}dx= \pi^{-s} \zeta(s)$$


