The Riemann zeta function and tuning: Difference between revisions

Line 279:

<math>\displaystyle{\left|\zeta\left(\frac{1}{2} + it\right)\right|^2 \cdot \overline {\phi(t)}}</math>

is, up to a flip in sign, the Fourier transform of the unnormalized Harmonic Shannon Entropy for {{nowrap|''N'' {{=}} ∞}}~~</math>~~, where φ(''t'') is the characteristic function (aka Fourier transform) of the spreading distribution and {{overline|φ(''t'')}} denotes complex conjugation.

is, up to a flip in sign, the Fourier transform of the unnormalized Harmonic Shannon Entropy for {{nowrap|''N'' {{=}} ∞}}, where φ(''t'') is the characteristic function (aka Fourier transform) of the spreading distribution and {{overline|φ(''t'')}} 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. In particular, when the spreading distribution is a Gaussian, the characteristic function is also a Gaussian.

@@ Line 279: / Line 279: @@
 <math>\displaystyle{\left|\zeta\left(\frac{1}{2} + it\right)\right|^2 \cdot \overline {\phi(t)}}</math>
-is, up to a flip in sign, the Fourier transform of the unnormalized Harmonic Shannon Entropy for {{nowrap|''N'' {{=}} &infin;}}</math>, where &phi;(''t'') is the characteristic function (aka Fourier transform) of the spreading distribution and {{overline|&phi;(''t'')}} denotes complex conjugation.
+is, up to a flip in sign, the Fourier transform of the unnormalized Harmonic Shannon Entropy for {{nowrap|''N'' {{=}} &infin;}}, where &phi;(''t'') is the characteristic function (aka Fourier transform) of the spreading distribution and {{overline|&phi;(''t'')}} 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. In particular, when the spreading distribution is a Gaussian, the characteristic function is also a Gaussian.