Derivative of Riemann Zeta Function

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Theorem

The derivative of the Riemann zeta function is:

$\ds \map {\zeta'} z = \frac {\d \zeta} {\d z} = -\sum_{n \mathop = 2}^\infty \frac {\map \ln n} {n^z}$


Proof

\(\ds \frac {\d \zeta} {\d z}\) \(=\) \(\ds \map {\frac \d {\d z} } {\sum_{n \mathop = 1}^\infty n^{-z} }\) Definition of Riemann Zeta Function
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \map {\frac \d {\d z} } {n^{-z} }\) Sum Rule for Derivatives/General Result
\(\ds \) \(=\) \(\ds \sum_{n \mathop = 1}^\infty \paren {-\map \ln n n^{-z} }\) Derivative of Exponential Function
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 1}^\infty \frac {\map \ln n} {n^z}\) Exponent Combination Laws/Negative Power
\(\ds \) \(=\) \(\ds -\sum_{n \mathop = 2}^\infty \frac {\map \ln n} {n^z}\) Natural Logarithm of 1 is 0

$\blacksquare$


Also see