Riemann zeta function/Appendix: Difference between revisions

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 === 1b. Conversion factor for removing primes ===
-{{todo|derive this|inline=1}}
+Consider the expression <math>\left|1 - p^{-s} \right|</math>, where <math>s = \sigma + it</math>. Here,
+<math>\displaystyle
+	p^{-s} = p^{-\sigma} p^{-it} = p^{-\sigma} e^{-it \ln p}
+</math>,
+so,
+<math>\displaystyle
+	\left|1 - p^{-s} \right|^2 = \left|1 - p^{-\sigma} e^{-it \ln p} \right|^2 = \left(1 - p^{-\sigma} e^{-it \ln p} \right) \left(1 - p^{-\sigma} e^{it \ln p} \right)
+</math>,
+which evaluates to:
+<math>\displaystyle
++ p^{-2\sigma} - 2p^{-\sigma} \frac{e^{-it \ln p} + e^{it \ln p}}{2}
+</math>,
+and by the definition of cosine in terms of complex exponentials,
+<math>\displaystyle
+	 \left|1 - p^{-s} \right| = \sqrt{1 + p^{-2\sigma} - cos(t \ln p)}
+</math>.
 == 2. Z function and Riemann-Siegel theta function ==