The Riemann zeta function and tuning/Appendix: Difference between revisions

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Below proceeds a mathematically rigorous exposition of the Z function and theta function, cut from Gene Ward Smith's derivation for the sake of clarifying the actual steps taken.

In order to define the Z function, we need first to define the {{w|Riemann–Siegel theta function}}, and in order to do that, we first need to define the [http://mathworld.wolfram.com/LogGammaFunction.html Log Gamma function]. This is not defined as the natural log of the Gamma function since that has a more complicated branch cut structure; instead, the principal branch of the Log Gamma function is defined as having a branch cut along the negative real axis, and is given by the series

where γ is the {{w|Euler–Mascheroni constant}}. We now may define the Riemann–Siegel theta function as

Another approach is to substitute {{nowrap|''z'' {{=}} {{sfrac|1 + 2''it''|4}}}} into the series for Log Gamma and take the imaginary part, this yields

Since the arctangent function is holomorphic in the strip with imaginary part between −1 and 1, it follows from the above formula, or arguing from the previous one, that θ is holomorphic in the strip with imaginary part between −{{frac|1|2}} and {{frac|1|2}}. It may be described for real arguments as an odd real analytic function of ''x'', increasing when {{nowrap|{{abs|''x''}} > 6.29}}. Plots of it may be studied by use of the Wolfram [http://functions.wolfram.com/webMathematica/FunctionPlotting.jsp?name=RiemannSiegelTheta online function plotter].

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 {{nowrap|ζ({{frac|1|2}} + ''ig'')}} is real. This implies that exp(''i''θ(''g'')) is real, so that {{frac|θ(''g'')|π}} is an integer. Theta has an {{w|asymptotic expansion}}

From this we may deduce that {{nowrap|{{sfrac|θ(''t'')|π}} ≈ ''r'' ln(''r'') − ''r'' − {{sfrac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''t''|2π}} {{=}} {{sfrac|''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'', which is {{nowrap|{{sfrac|1200|ln(2)}} {{=}} 1731.234{{cent}}}}.

The fact that ''x'' is slightly greater than 12 means 12 has an overall sharp quality. We may also find this out by looking at the value we computed for {{nowrap|θ(2π''r'') / π}}, which was 31.927. Then {{nowrap|32 − 31.927 {{=}} 0.0726}}, which is positive but not too large; this is the second black magic formula, evaluating the nature of an edo ''x'' by computing {{nowrap|⌊''r'' ln(''r'') − ''r'' + {{frac|3|8}}⌋ − ''r'' ln(''r'') + ''r'' + {{frac|1|8}}}}, where {{nowrap|''r'' {{=}} {{sfrac|''x''|ln(2)}}}}. This works more often than not on the clearcut cases, but when ''x'' is extreme it may not; 49 is very sharp in tendency, for example, but this method calls it as flat; similarly it counts 45 as sharp.

There are various approaches to the question of computing the zeta function, but perhaps the simplest is the use of the {{w|Dirichlet eta function}} which was introduced to mathematics by {{w|Johann Peter Gustav Lejeune Dirichlet}}, who despite his name was a German and the brother-in-law of {{w|Felix Mendelssohn}}.

The zeta function has a [http://mathworld.wolfram.com/SimplePole.html simple pole] at {{nowrap|''z'' {{=}} 1}} which forms a barrier against continuing it with its {{w|Euler product}} or {{w|Dirichlet series}} representation. We could subtract off the pole, or multiply by a factor of {{nowrap|''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 {{nowrap|1 − 2^{1 − ''z''}}}, leading to the eta function:

The Dirichlet series for the zeta function is absolutely convergent when {{nowrap|''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 {{nowrap|1 + {{sfrac|2π''i''|ln(2)}}x}} corresponding to pure octave divisions along the line {{nowrap|''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 {{w|Euler summation}}.