Riemann zeta function: Difference between revisions

Line 87:

<math>\displaystyle F_s(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}</math>

This new function has the property that {{nowrap|Fs(''x'') {{=}} Fs(0) − Es(''x'')}}, so that all we have done is flip the sign of Es(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function~~, which~~ can be expressed in terms of ~~the real part of the logarithm of the~~ {{w|~~Riemann zeta function~~}}:

This new function has the property that {{nowrap|Fs(''x'') {{=}} Fs(0) − Es(''x'')}}, so that all we have done is flip the sign of Es(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum.

Of more interest is the fact that it is a known mathematical function.

The logarithm of the {{w|Riemann zeta function}} function can be expressed in terms of a {{w|Dirichlet series}} involving the Von Mangoldt function:

<math>\displaystyle ~~F_s~~(x) = \~~mathrm~~{Re} \~~left( \ln \zeta \left(s +~~ \frac{2 \~~pi i~~}{\ln 2}x \~~right)~~ \~~right)~~</math>

<math>\displaystyle

\ln \zeta(s)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\ln(n)}\,\frac{1}{n^s}

</math>

{{~~Todo|expand|inline~~=~~1|text~~=~~Make it clear how Fₛ~~(x) ~~relates to the~~ zeta ~~function. Due to the sudden appearance of the natural logarithm and the imaginary unit i, this appears to have to do with complex exponentials~~ (i~~.e. those found in the denominator of the terms of zeta when the input is complex), but it would be more approachable if the precise derivation was laid out here.~~}}

We can rewrite the cosine term in <math>F_s(x)</math> using the real part of an exponential:

<math>\displaystyle

\cos(2 \pi x \log_2 n)

= \mathrm{Re}\left( \exp{(2 \pi i x \log_2 n)} \right)

= \mathrm{Re}\left( n^{\frac{2 \pi i}{\ln 2} x} \right)

</math>

Substituting back into <math>F_s(x)</math> gives:

<math>\displaystyle

F_s(x)

= \mathrm{Re} \left( \sum_{n=2}^\infty \frac{\Lambda(n)}{\ln n} \frac{n^{\frac{2 \pi i}{\ln 2} x}}{n^s} \right)

= \mathrm{Re} \left( \ln \zeta \left(s + \frac{2 \pi i}{\ln 2}x \right) \right)

</math>

If we take exponentials of both sides, then

@@ Line 87: / Line 87: @@
 <math>\displaystyle F_s(x) = \sum_{n \geq 1} \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}</math>
-This new function has the property that {{nowrap|F<sub>s</sub>(''x'') {{=}} F<sub>s</sub>(0) − E<sub>s</sub>(''x'')}}, so that all we have done is flip the sign of E<sub>s</sub>(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum. Of more interest is the fact that it is a known mathematical function, which can be expressed in terms of the real part of the logarithm of the {{w|Riemann zeta function}}:
+This new function has the property that {{nowrap|F<sub>s</sub>(''x'') {{=}} F<sub>s</sub>(0) − E<sub>s</sub>(''x'')}}, so that all we have done is flip the sign of E<sub>s</sub>(''x'') and offset it vertically. This now increases to a maximum value for low errors, rather than declining to a minimum.
+Of more interest is the fact that it is a known mathematical function.
+The logarithm of the {{w|Riemann zeta function}} function can be expressed in terms of a {{w|Dirichlet series}} involving the Von Mangoldt function:
-<math>\displaystyle F_s(x) = \mathrm{Re} \left( \ln \zeta \left(s + \frac{2 \pi i}{\ln 2}x \right) \right)</math>
+<math>\displaystyle
+	\ln \zeta(s)=\sum_{n=2}^\infty \frac{\Lambda(n)}{\ln(n)}\,\frac{1}{n^s}
+</math>
-{{Todo|expand|inline=1|text=Make it clear how Fₛ(x) relates to the zeta function. Due to the sudden appearance of the natural logarithm and the imaginary unit i, this appears to have to do with complex exponentials (i.e. those found in the denominator of the terms of zeta when the input is complex), but it would be more approachable if the precise derivation was laid out here.}}
+We can rewrite the cosine term in <math>F_s(x)</math> using the real part of an exponential:
+<math>\displaystyle
+	\cos(2 \pi x \log_2 n)
+	 = \mathrm{Re}\left( \exp{(2 \pi i x \log_2 n)} \right)
+	 = \mathrm{Re}\left( n^{\frac{2 \pi i}{\ln 2} x} \right)
+</math>
+Substituting back into <math>F_s(x)</math> gives:
+<math>\displaystyle
+	F_s(x)
+	 = \mathrm{Re} \left( \sum_{n=2}^\infty \frac{\Lambda(n)}{\ln n} \frac{n^{\frac{2 \pi i}{\ln 2} x}}{n^s} \right)
+	 = \mathrm{Re} \left( \ln \zeta \left(s + \frac{2 \pi i}{\ln 2}x \right) \right)
+</math>
 If we take exponentials of both sides, then