Graph of $1/\zeta(n)$ where $n$ runs through positive real numbers .

Question

Can we get a graph of $(1/\zeta(n))$ for $n$ belonging real numbers ?

I know that as

$n \rightarrow \infty$

$\zeta \rightarrow 1$

i.e. it is asymptotic to 1

but what is an exact graph looks like ?

( Only root of the equation is $n=1$ in the given domain )

Answer 1

The reciprocal of the Riemann zeta function has been plotted, for example here, at the bottom of the page, e.g., $\frac{1}{\zeta(x)}$ for real $x\ge 1$.

The linked article on the Riemann zeta function is interesting in itself and has several other nice graphs included.

Answer 2

For $s > 1$ and by analytic continuation for $s> 0$ $$\zeta(s) = \frac1{s-1}+\sum_{n=1}^\infty (n^{-s}-\int_n^{n+1} x^{-s}dx)$$

Thus for any $N \in \Bbb{Z}_{\ge 2}$ and for all $s > 0$ $$ |\zeta(s)-\sum_{n=1}^N n^{-s}-\frac{(N+1)^{-s}}{s-1}| =|\sum_{n=N+1}^\infty \int_n^{n+1} \int_n^x s t^{-s-1}dtdx| \le \int_{N+1}^\infty sx^{-s-1}dx \le (N+1)^{-s}$$

This is enough to plot $1/\zeta(s)$ for $s > \epsilon$