# Convergence of $\sum_{n=1}^{\infty} \frac{1}{n^z}$

Let us consider $z\in \mathbb C$; what is the condition on modulus of z in order that $$\sum_{n=1}^{\infty} \frac{1}{n^z}$$ the series (zeta function?) converges? For example, if $|z|=1$, the series diverges? Thank you very much.

• Answer to this can be googled in a second. – Harto Saarinen Oct 21 '14 at 16:45
• Is $|z|\geq 1$? – Mark Oct 21 '14 at 16:52

The Riemann zeta function or Euler–Riemann zeta function, $ζ(s)$, is a function of a complex variable s that analytically continues the sum of the infinite series $$\sum_{n=1}^\infty\frac{1}{n^s}$$ which converges when the real part of s is greater than 1