Let $$M(N)=\sum_{n=1}^{N}\mu(n)$$
be a Mertens function.
According to the fact that $$\frac{1}{\zeta(s)}=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}$$
And fact that $\zeta(0)=-\frac{1}{2}$
Then i conclude that $$\lim_{N\to\infty}M(N)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{0}}=\frac{1}{\zeta(0)}=-2$$
Hence $M(N)$ is bounded ... But as we know it is not bounded.
Could you spot the mistake?
Regards.