The Riemann zeta function and tuning/Appendix

< The Riemann zeta function and tuning

Euler product expression for the zeta function

Starting with the definition of the zeta function:

[math]\displaystyle{ \displaystyle \zeta(s) = \sum_{n = 1}^\infty \frac{1}{n^s} }[/math],

by the Fundamental Theorem of Arithmetic, and for [math]\displaystyle{ \mathrm{Re}(s) \gt 1 }[/math], where the series is absolutely convergent, it can be rearranged into a sum over prime factorizations:

[math]\displaystyle{ \displaystyle \zeta(s) = \sum_{r_k = 0}^\infty \frac{1}{\left(\prod_{k = 1}^\infty {p_k}^{r_k} \right)^s} = \sum_{r_k = 0}^\infty \prod_{k = 1}^\infty \frac{1}{{p_k}^{s r_k}} }[/math],

where [math]\displaystyle{ p_k }[/math] is the k-th prime.

The sum on the outside is really an infinite product of sum operators for each k, and so:

[math]\displaystyle{ \displaystyle \sum_{r_k = 0}^\infty \prod_{k = 1}^\infty \frac{1}{{p_k}^{s r_k}} = \prod_{k = 1}^\infty \sum_{r = 0}^\infty \frac{1}{\left({p_k}^s \right)^r} }[/math],

which is a geometric series, and given that [math]\displaystyle{ {p_k}^s }[/math] is greater than 1 for all k when [math]\displaystyle{ \mathrm{Re}(s) \gt 1 }[/math],

[math]\displaystyle{ \displaystyle \zeta(s) = \prod_{k = 1}^\infty \frac{1}{1 - {p_k}^{-s}}, }[/math]

which is the Euler product formula we desired.

Dirichlet series for the von Mangoldt function

Start with the Euler Product for \(\zeta(s)\). For \(\mathrm{Re}(s) > 1\), the Riemann zeta function can be expressed as:

[math]\displaystyle{ \displaystyle \zeta(s) = \prod_{p \text{ prime}} \left(1 - p^{-s}\right)^{-1} }[/math].

Take the natural logarithm to convert the product into a sum:

[math]\displaystyle{ \displaystyle \ln \zeta(s) = \sum_{p \text{ prime}} - \ln \left( 1 - p^{-s} \right) }[/math].

For \( |x| < 1 \), we have the well-known series:

[math]\displaystyle{ \displaystyle -\ln(1 - x) = \sum_{k=1}^{\infty} \frac{x^k}{k} }[/math].

Substitute \( x = p^{-s} \). When \( \mathrm{Re}(s) > 0 \), we have \( |p^{-s}| = p^{-\mathrm{Re}(s)} < 1 \) for any \( p \geq 2 \), ensuring convergence:

[math]\displaystyle{ \displaystyle -\ln\left(1 - p^{-s}\right) = \sum_{k=1}^{\infty} \frac{\left(p^{-s}\right)^k}{k} }[/math].

Simplify the expression. Noting that \( \left(p^{-s}\right)^k = \left(p^k\right)^{-s} \):

[math]\displaystyle{ \displaystyle \sum_{k=1}^{\infty} \frac{\left(p^{-s}\right)^k}{k} = \sum_{k=1}^\infty \frac{1}{k} \left(p^k\right)^{-s} }[/math].

We thus have:

[math]\displaystyle{ \displaystyle - \ln\left(1 - p^{-s}\right) = \sum_{k=1}^\infty \frac{1}{k} \left(p^k\right)^{-s} }[/math]

for all \( p \geq 2 \) and \( \mathrm{Re}(s) > 0 \).

We can now substitute this back in our earlier sum:

[math]\displaystyle{ \displaystyle \ln \zeta(s) = \sum_{p \text{ prime}} - \ln \left( 1 - p^{-s} \right) = \sum_{p \text{ prime}} \sum_{k=1}^\infty \frac{1}{k} \left(p^k\right)^{-s} }[/math].

The von Mangoldt function \(\Lambda(n)\) identifies prime powers:

[math]\displaystyle{ \displaystyle \Lambda(n) = \begin{cases} \ln p & \text{if } n = p^k \text{ for prime } p \text{ and } k \geq 1 \text{,} \\ 0 & \text{otherwise.} \end{cases} }[/math]

Reindex the double sum. Each term in the double sum corresponds to a prime power \(n = p^k\). For any integer \(n \geq 2\), \(n\) contributes to the sum if and only if it's a prime power.

For \(n = p^k\):

[math]\displaystyle{ \displaystyle \frac{\Lambda(n)}{\ln n} = \frac{\ln p}{k \ln p} = \frac{1}{k} }[/math].

Therefore:

[math]\displaystyle{ \displaystyle \ln \zeta(s) = \sum_{n=2}^\infty \frac{\Lambda(n)}{\ln n} \cdot \frac{1}{n^s} }[/math],

which is valid for \(\mathrm{Re}(s) > 1\), with the sum starting at \(n=2\) because \(\Lambda(1) = 0\).

Conversion factor for removing primes

Consider the expression [math]\displaystyle{ \left|1 - p^{-s} \right| }[/math], where [math]\displaystyle{ s = \sigma + it }[/math]. Here,

[math]\displaystyle{ \displaystyle p^{-s} = p^{-\sigma} p^{-it} = p^{-\sigma} e^{-it \ln p} }[/math],

so,

[math]\displaystyle{ \displaystyle \left|1 - p^{-s} \right|^2 = \left|1 - p^{-\sigma} e^{-it \ln p} \right|^2 = \left(1 - p^{-\sigma} e^{-it \ln p} \right) \left(1 - p^{-\sigma} e^{it \ln p} \right) }[/math],

which evaluates to:

[math]\displaystyle{ \displaystyle 1 + p^{-2\sigma} - 2p^{-\sigma} \frac{e^{-it \ln p} + e^{it \ln p}}{2} }[/math],

and by the definition of cosine in terms of complex exponentials,

[math]\displaystyle{ \displaystyle \left|1 - p^{-s} \right| = \sqrt{1 + p^{-2\sigma} - 2p^{-\sigma} \cos(t \ln p)} }[/math].

Z function and Riemann-Siegel theta function

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 Riemann–Siegel theta function, and in order to do that, we first need to define the 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

[math]\displaystyle{ \displaystyle \Upsilon(z) = -\gamma z - \ln z + \sum_{k=1}^\infty \left(\frac{z}{k} - \ln\left(1 + \frac{z}{k}\right)\right) }[/math]

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

[math]\displaystyle{ \displaystyle \theta(z) = \frac{\Upsilon\left(\frac{1 + 2 i z}{4}\right) - \Upsilon\left(\frac{1 - 2 i z}{4}\right)}{2 i} - \frac{\ln(\pi)}{2} z }[/math]

Another approach is to substitute z = ⁠1 + 2it/4⁠ into the series for Log Gamma and take the imaginary part, this yields

[math]\displaystyle{ \displaystyle \theta(t) = -\frac{\gamma + \log \pi}{2}t - \arctan 2t + \sum_{n=1}^\infty \left(\frac{t}{2n} - \arctan\left(\frac{2t}{4n+1}\right)\right) }[/math]

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 −1⁄2 and 1⁄2. It may be described for real arguments as an odd real analytic function of x, increasing when |x| > 6.29. Plots of it may be studied by use of the Wolfram online function plotter.

Black magic formulas

When 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 asymptotic expansion

[math]\displaystyle{ \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 = ⁠1/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 ⁠1200/ln(2)⁠ = 1731.234 ¢.

Recall that Gram points near to pure-octave edos, where x is an integer, can be expected to correspond to peak values of |ζ| = |Z|. We can find these Gram points by Newton's method applied to the above formula. If r = ⁠x/ln(2)⁠, and if n = ⌊r ln(r) − r + 3⁄8⌋ is the nearest integer to ⁠θ(2πr)/π⁠, then we may set r⁺ = ⁠r + n + 1⁄8/ln(r)⁠. This is the first iteration of Newton's method, which we may repeat if we like, but in fact no more than one iteration is really required. This is the first black magic formula, giving an adjusted "Gram" tuning from the orginal one.

For an example, consider x = 12, so that r = ⁠12/ln(2)⁠ = 17.312. Then r ln(r) − r − ⁠1/8⁠ = 31.927, which rounded to the nearest integer is 32, so n = 32. Then ⁠r + n + 1⁄8/ln(r)⁠ = 17.338, corresponding to x = 12.0176, which means a single step is 99.853 cents and the octave is tempered to twelve of these, which is 1198.238 cents.

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 θ(2πr) / π, which was 31.927. Then 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 ⌊r ln(r) − r + 3⁄8⌋ − r ln(r) + r + 1⁄8, where r = ⁠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.

Computing zeta

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

The zeta function has a simple pole at z = 1 which forms a barrier against continuing it with its Euler product or 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]\displaystyle{ \displaystyle \eta(z) = \left(1-2^{1-z}\right)\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]

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 + ⁠2πi/ln(2)⁠x 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 Euler summation.