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 |
$\blacksquare$