The Riemann zeta function and tuning: Difference between revisions

Line 236:

The Dirichlet series for the zeta function is absolutely convergent when s>1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2πix/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying [http://en.wikipedia.org/wiki/Euler_summation Euler summation].

=Relationship to Harmonic Entropy=

The expression

<math>|\zeta(0.5+it)|^2 \cdot \overline {\phi(t)}</math>

is the Fourier transform of the unnormalized Harmonic Shannon Entropy for <math>N=\infty</math>, where <math>\phi(t)</math> is the characteristic function of the spreading distribution and <math>\overline {\phi(t)}</math> denotes complex conjugation.

Note that in the most common case where the spreading distribution is symmetric (as in the case of the Gaussian and Laplace distributions), the characteristic function is purely real and hence the conjugate is unnecessary.

More can be found at the page on [[Harmonic_Entropy#Extending_HE_to_.5Bmath.5DN.3D.5Cinfty.5B.2Fmath.5D:_zeta-HE|Harmonic Entropy]], including a generalization to Renyi entropy for arbitrary <math>a</math>.

=Links=

@@ Line 236: / Line 236: @@
 The Dirichlet series for the zeta function is absolutely convergent when s&gt;1, justifying the rearrangement of terms leading to the alternating series for eta, which converges conditionally in the critical strip. The extra factor introduces zeros of the eta function at the points 1 + 2πix/ln(2) corresponding to pure octave divisions along the line s=1, but no other zeros, and in particular none in the critical strip and along the critical line. The convergence of the alternating series can be greatly accelerated by applying [http://en.wikipedia.org/wiki/Euler_summation Euler summation].
+=Relationship to Harmonic Entropy=
+The expression
+<math>|\zeta(0.5+it)|^2 \cdot \overline {\phi(t)}</math>
+is the Fourier transform of the unnormalized Harmonic Shannon Entropy for <math>N=\infty</math>, where <math>\phi(t)</math> is the characteristic function of the spreading distribution and <math>\overline {\phi(t)}</math> denotes complex conjugation.
+Note that in the most common case where the spreading distribution is symmetric (as in the case of the Gaussian and Laplace distributions), the characteristic function is purely real and hence the conjugate is unnecessary.
+More can be found at the page on [[Harmonic_Entropy#Extending_HE_to_.5Bmath.5DN.3D.5Cinfty.5B.2Fmath.5D:_zeta-HE|Harmonic Entropy]], including a generalization to Renyi entropy for arbitrary <math>a</math>.
 =Links=