Riemann zeta function: Difference between revisions
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=== Relationship to harmonic entropy === | === Relationship to harmonic entropy === | ||
The expression | The expression | ||
<math>|\zeta(0.5+it)|^2 \cdot \overline {\phi(t)}</math> | <math>\displaystyle{|\zeta(0.5+it)|^2 \cdot \overline {\phi(t)}}</math> | ||
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 (aka Fourier transform) 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 (aka Fourier transform) of the spreading distribution and <math>\overline {\phi(t)}</math> denotes complex conjugation. | ||
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The [http://mathworld.wolfram.com/EulerProduct.html Euler product] for the Riemann zeta function is | The [http://mathworld.wolfram.com/EulerProduct.html Euler product] for the Riemann zeta function is | ||
<math>\zeta(s) = \prod_p (1 - p^{-s})^{-1}</math> | <math>\displaystyle{ | ||
\zeta(s) = \prod_p (1 - p^{-s})^{-1} | |||
}</math> | |||
where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, (1-2^(-s))ζ(s) is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3. | where the product is over all primes p. The product converges for values of s with real part greater than or equal to one, except for s=1 where it diverges to infinity. We may remove a finite list of primes from consideration by multiplying ζ(s) by the corresponding factors (1-p^(-s)) for each prime p we wish to remove. After we have done this, the smallest prime remaining will dominate peak values for s with large real part, and as before we can track these peaks backwards and, by analytical continuation, into the critical strip. In particular if we remove the prime 2, (1-2^(-s))ζ(s) is now dominated by 3, and the large peak values occur near equal divisions of the "tritave", ie 3. | ||
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Along the critical line, |1 - p^(-1/2-i t)| may be written | Along the critical line, |1 - p^(-1/2-i t)| may be written | ||
<math>\sqrt{1 + \frac{1}{p} - \frac{2 \cos(\ln p\ t)}{\sqrt{p}}}</math> | <math>\displaystyle{ | ||
\sqrt{1 + \frac{1}{p} - \frac{2 \cos(\ln p\ t)}{\sqrt{p}}} | |||
}</math> | |||
Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime p removed from consideration. Zeta peak and zeta integral tunings may then be found as before. | Multiplying the Z-function by this factor of adjustment gives a Z-function with the prime p removed from consideration. Zeta peak and zeta integral tunings may then be found as before. | ||
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When [[Gene_Ward_Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a point on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]] | When [[Gene_Ward_Smith|Gene Smith]] discovered these formulas in the 70s, he thought of them as "black magic" formulas not because of any aura of evil, but because they seemed mysteriously to give you something for next to nothing. They are based on Gram points and the Riemann-Siegel theta function θ(t). Recall that a Gram point is a point on the critical line where ζ(1/2 + ig) is real. This implies that exp(iθ(g)) is real, so that θ(g)/π is an integer. Theta has an [[Wikipedia:asymptotic expansion|asymptotic expansion]] | ||
<math>\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots</math> | <math>\displaystyle{ | ||
\theta(t) \sim \frac{t}{2}\log \frac{t}{2\pi} - \frac{t}{2} - \frac{\pi}{8}+\frac{1}{48t}+ \frac{7}{5760t^3}+\cdots | |||
}</math> | |||
From this we may deduce that θ(t)/π ≈ r ln(r) - r - 1/8, where r = t/2π = x/ln(2); hence while x is the number of equal steps to an octave, r is the number of equal steps to an "e-tave", meaning the interval of e, 1200/ln(2) = 1731.234 cents. | From this we may deduce that θ(t)/π ≈ r ln(r) - r - 1/8, where r = t/2π = x/ln(2); hence while x is the number of equal steps to an octave, r is the number of equal steps to an "e-tave", meaning the interval of e, 1200/ln(2) = 1731.234 cents. | ||
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The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at z=1 which forms a barrier against continuing it with its [[Wikipedia:Euler product|Euler product]] or [[Wikipedia:Dirichlet series|Dirichlet series]] representation. We could subtract off the pole, or multiply by a factor of (z-1), but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of (1-2^(1-z)), leading to the eta function: | The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at z=1 which forms a barrier against continuing it with its [[Wikipedia:Euler product|Euler product]] or [[Wikipedia:Dirichlet series|Dirichlet series]] representation. We could subtract off the pole, or multiply by a factor of (z-1), but at the expense of losing the character of a Dirichlet series or Euler product. A better method is to multiply by a factor of (1-2^(1-z)), leading to the eta function: | ||
<math>\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z} | <math>\displaystyle{\eta(z) = (1-2^{1-z})\zeta(z) = \sum_{n=1}^\infty (-1)^{n-1} n^{-z} | ||
= \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots</math> | = \frac{1}{1^z} - \frac{1}{2^z} + \frac{1}{3^z} - \frac{1}{4^z} + \cdots}</math> | ||
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 [[Wikipedia:Euler summation|Euler summation]]. | 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 [[Wikipedia:Euler summation|Euler summation]]. | ||