Can the Von-Mangoldt function and the Chebyshev function be defined for entire complex plane?

Question

Can the von-Mangoldt function and the Chebyshev function be defined for the entire complex plane ?

I assume so, but I had not seen the definition. Can anyone provide some links for this?

Thank you.

Answer 1

The von Mangoldt function can be generated at the pole $s=1$ of Riemann zeta with the formula: $$\Lambda(n,s)=\lim\limits_{s \rightarrow 1} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}$$

So you can instead of $s \rightarrow 1$ let $s$ be complex number so that you have: $s=a+ib$

$$\Lambda(n,a,b)=\lim\limits_{s \rightarrow a+ib} \zeta(s)\sum\limits_{d|n} \frac{\mu(d)}{d^{(s-1)}}$$