Maple Riemann Zeta Function does not output 0

Question

I tried using the Zeta function in Maple, though instead of using Zeta directly I used Sum:

sum(1/i^s,i=1)

The problem here though, is that if:

s := 0.5 + 3i

then
sum(1/i^s, i=1) = 1.011077905

This does not output 0, which is what it is supposed to be, proven by Riemann, so it eventually means that I have done something wrong. I would be grateful for some help,

Thanks.

Answer 1

I think that there is a confusion between $i$ (the index) and $i$ (the imaginary number) ! $$\zeta (s)= \sum_{n=1}^\infty \frac 1 {n^s}$$ and $$\zeta \left(\frac{1}{2}+3 i\right)\approx 0.532737 -0.0788965 i$$ What it seems it that you computed $$\sum_{n=1}^\infty \frac 1 {n^{\frac{1}{2}+3 n}}\approx 1.0110779057055353498$$