Does this formula correspond to a series representation of the Dirac delta function $\delta(x)$?

$$\sum_k e^{2i\pi kx} = \sum_m \delta(x-m)$$

Convergence in the sense of distributions

$$\lim_{N\to \infty,M(N)=0}\sum_{n=1}^N \frac{\mu(n)}{n} \sum_k e^{2i\pi kx/n} =\lim_{N\to \infty,M(N)=0}\sum_{n=1}^N \mu(n) \sum_n\delta(x-mn)$$ $$=\lim_{N\to \infty,M(N)=0}\sum_{l\ge 1}(\delta(x+l)+\delta(x-l))\sum_{d| l,d\le N} \mu(d) =\delta(x+1)+\delta(x-1)$$