Question about non trivial zeros of Riemann zeta function Riemann zeta function is $$\zeta(s)=\sum\frac{1}{n^s}$$ I read at wiki that the first nontrivial zero is located at $14.134725\ldots$ As far as I understand it means $$\zeta(s)=\sum\frac{1}{n^{0.5+i14.134725\ldots}}$$ Does it means $$\sum\frac{1}{n^{0.5+i14.134725\ldots}}$$ converges to $0$ for $n=1$ to infinity?

Is is not true. The sum $$\zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s}\tag1$$ converges only for $\Re(x)>1$. Nevertheless we can extend it on the whole complex plane (except $x=1$) so that it is holomorphic. Then, it means that the expansion of this function has a zero at $0.5+i14.134725...$
The false belief is that the $\zeta$ function is defined as $(1)$. It is defined as an expansion of this sum, not the sum itself.
