The Riemann zeta function and tuning: Difference between revisions
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<math>|\zeta(0.5+it)|^2 \cdot \overline {\phi(t)}</math> | <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. | is, up to a flip in sign, 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. | 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. | ||