= Logarithmic derivative
{title2=$f'/f$}
{wiki}
The <logarithmic derivative> of a <differentiable function> $f$ is $f'/f$ wherever $f\ne0$. For a nonvanishing <holomorphic function>, a local <holomorphic logarithm> differentiates to this expression even when no global logarithm exists. If a <meromorphic function> has local factorization $f(z)=(z-a)^mh(z)$ with $h(a)\ne0$, differentiation gives $f'/f=m/(z-a)+h'/h$. Its <residue> is therefore $m$, positive for a zero and negative for a <pole>; this explains the multiplicity count in the <argument principle>. For $\Re s>1$, the <Euler product> for the <Riemann zeta function> gives
$$
-\frac{\zeta'(s)}{\zeta(s)}
=\sum_{n\geq1}\frac{\Lambda(n)}{n^s}.
$$
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