How to compute $\lim_{s \to 1} (s-1) \frac{\zeta'(s)}{\zeta(s)} $ ?

Question

I wish to verify the conditions of a certain theorem to prove that the integral $$\int_{1}^{\infty} \frac{\psi(x) - x}{x^2} dx $$ converges. (Where $\psi(x) = \sum_{n\leq x} \Lambda (n) $, and $\Lambda$ is the von Mangoldt funtion.) To do this, I need to compute $$\lim_{s \to 1} (s-1) \frac{\zeta'(s)}{\zeta(s)} ,$$ but I am not entirely sure how to do so. I guess one could use the functional equation for the zeta function, but I suppose I'm missing something. Could you please help me in the right direction?

Answer 1

The Riemann zeta function has a simple pole at $s = 1$ with the Laurent expansion

$$ \zeta(s) = \frac 1{s-1} + O(1) $$

It follows that $$ \zeta'(s) = -\frac 1{(s-1)^2} + O(1) $$ for $s \to 1$ and therefore $$ \lim_{s \to 1} (s-1) \frac{\zeta'(s)}{\zeta(s)} = -1 \quad . $$

Generally, if $f$ is meromorphic in a neighborhood of $z_0$ and has a zero of order $k$ (or a pole of order $-k$) at $z_0$, then

$$ \lim_{z \to z_0} (z-z_0) \frac{f'(z)}{f(z)} = k \quad . $$