How to find $\zeta(s)$ for $\text{Re}(s)\leq 1$?

Question

I was trying to build animation to show How $\zeta(1/2+it)$ would look as $t$ changes. If the $\text{Re}(s)$ were greater than $1$, I could simply use the definition of the zeta function. $$\zeta(s)=\sum_{n=0}^\infty \frac{1}{n^s}$$

But as the above sum doesn't converge for $\text{Re}(s)\leq 1$, I'm in trouble. I don't know if I can use analytic continuation for it. So I'm asking if there is some trivial way to find $\zeta(1/2+it)$ for $\text{Re}(s)\leq 1$.

If I were you, I'd hurry an have a look at the only answer before it gets deleted. It answers your question perfecty. — G. Fougeron, Jun 02 '21 at 13:24
Look at this https://math.stackexchange.com/questions/437883/what-is-the-analytic-continuation-of-the-riemann-zeta-function — David, Jun 02 '21 at 23:35

score 1 · Answer 1 · edited Jun 02 '21 at 14:39

1

You can use the Dirichlet Eta function:

$$\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s}$$

as this is valid for $s\gt0$ (so includes all of the critical strip), and also:

$$\eta(s)=(1-2^{1-s})\zeta(s)$$

and so when $\eta(s)$ is a zero, so is $\zeta(s)$.

edited Jun 02 '21 at 14:39

vitamin d

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answered Jun 02 '21 at 13:27

JMP

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It should be $\eta(s)=(1-2^{1-s})\zeta(2)$. – Young Kindaichi Jun 02 '21 at 14:27

How to find $\zeta(s)$ for $\text{Re}(s)\leq 1$?

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