The Riemann zeta function and tuning: Difference between revisions

Line 25:

<math>E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}</math>

For any fixed s > 1 this gives a [http://en.wikipedia.org/wiki/Analytic_function real analytic function] defined for all x, and hence with all the smoothness properties we could desire. We can ~~define~~ essentially the same function ~~by subtracting it from E_s(1/2)/2~~:

For any fixed s > 1 this gives a [http://en.wikipedia.org/wiki/Analytic_function real analytic function] defined for all x, and hence with all the smoothness properties we could desire.

We can clean up this definition to get essentially the same function:

<math>F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}</math>

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 [http://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function]:

This new function has the property that <math>F_s(x) = F_s(0) - E_s(x)</math>, so that all we have done is flip the sign of <math>E_s(x)</math> 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 [http://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function]:

@@ Line 25: / Line 25: @@
 <math>E_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{1 - \cos(2 \pi x \log_2 n)}{n^s}</math>
-For any fixed s &gt; 1 this gives a [http://en.wikipedia.org/wiki/Analytic_function real analytic function] defined for all x, and hence with all the smoothness properties we could desire. We can define essentially the same function by subtracting it from E_s(1/2)/2:
+For any fixed s &gt; 1 this gives a [http://en.wikipedia.org/wiki/Analytic_function real analytic function] defined for all x, and hence with all the smoothness properties we could desire.
+We can clean up this definition to get essentially the same function:
 <math>F_s(x) = \sum_1^\infty \frac{\Lambda(n)}{\ln n} \frac{\cos(2 \pi x \log_2 n)}{n^s}</math>
-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 [http://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function]:
+This new function has the property that <math>F_s(x) = F_s(0) - E_s(x)</math>, so that all we have done is flip the sign of <math>E_s(x)</math> 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 [http://en.wikipedia.org/wiki/Riemann_zeta_function Riemann zeta function]:
 <math>F_s(x) = \Re \ln \zeta(s + 2 \pi i x/\ln 2)</math>