The Riemann Zeta function and n-et's

Let z = s + i*t be a complex number with real part s and imaginary
part t. If s>1 we can define a function

zeta(z) = sum_n n^(-z),

where n runs over the positive integers. This function analytically
continues over the whole complex plane, giving a meromorphic function
with a single simple pole at z=1. The sum is absolutely convergent
for s>1, and we may pass in this case to a corresponding infinite
product

zeta(z)= prod_p (1 - p^(-z))^(-1),

where the product is taken over all primes p. This is the key to the
zeta function--it encodes information about the primes. If we take
logarithms, we get

ln(zeta(z))= sum_p -ln(1-p^(-z)) = sum_p ln(1+p^z + p^(2z)+p^3z+...)

The real part of this is the log of the absolute value

Re(ln(zeta(z)) = ln(|zeta(z)|) = sum_p ln(|1-p^(-z)|).

If we fix s and let t vary, then each term of the above sum becomes a
periodic function of t with period 2 pi / ln(p). If we look at the
first-order approximation, we have

ln(1+p^(-z)) ~ p^(-z) = p^(-s) (cos(ln(p) t) - i sin(ln(p) t)),

so that the real part of ln(zeta(z)) is approximately

sum_p p^(-s) cos(ln(p) t)

If we rescale by setting t = 2 pi n /ln(2), we get a sum weighted by
factors of 1/p^s of cosine functions with periods of log_2(p). If n
is a real number near a good et (e.g. n=12.018) then for the smallest
primes, which have the highest weight, we will simultaneously be near
a peak value of several of these cosine functions. The functions we
are actually summing are also periodic but not exactly cosines; they
in fact are an improvement, which take into account the powers of the
prime p and so sharpen the peaks near their maximum, particularly for
the smaller primes where we are especially concerned to accurately
represent prime powers.

Since the infinite product converges when s>=1, we are justified in
taking values of
t=2 pi n /ln(2) which give relatively high maximums for the fixed
value of s as representing good scale divisions. As s goes to
infinity, this becomes increasingly a matter of finding good values
for the fifth, which we can much more easily accomplish via a
continued fraction. More interesting are the cases with smaller
values of s, as these give more weight to the larger primes and prime
powers.

It is a little harder to justify taking s into the critical strip
between 0 and 1. However, the fact of analytic continuation and the
Riemann-Siegel formula helps to make sense of this also, at least up
to the critical line s=1/2; this is particularly the case if the
Riemann hypothesis is true, in which case ln(zeta(z)) is analytic in
the strip up to the critical line. We have a functional equation
relating values of s to values of 1/2-s, so we don't want to push it
past the critical line in any case. When we continue past s=1, we may
follow a line of steepest ascent up to a high value of absolute value
for zeta, and particularly when we get a good et which does well for
a number of primes we might expect to reach a high maximum.

We may adjust the zeta function along the critical line by setting

Z(t) = zeta(1/2 + i t) pi^(-i t) Gamma(1/4 + i t/2)/|Gamma(1/4 + i
t/2)|

this makes Z into a real function of the real variable t, whose
absolute value is the same as that of zeta(1/2+i t). Here Gamma is
the Gamma function, where Gamma(z) = (z-1)! for complex z.

We have an approximate formula for Z,

Z(t) ~ 2 sum_{n <= L} cos(t ln(n) - theta(t))/sqrt(n)

where L = sqrt(t / 2 pi) and theta has an asymptotic expression

theta(t) ~~ (t/2) ln(t / 2 pi) - t/2 - pi/8 + 1/ (48 t) + 7/(5760
t^3) + ...

which is extremely accurate for our purposes.

We see that if theta(t) is a multiple of pi, we have again a weighted
sum of cosines. Hence we are justified at looking at maximums of Z(t)
particularly at "Gram points", which are points where theta(t) is an
integer multiple of pi and so cos(theta(t))=+-1, sin(theta(t))=0. If
we divide theta by pi and set r = t / 2 pi, we get

f(r) = theta(t / 2 pi)/pi ~ r ln(r) - r - 1/8

and Gram points will be integer values of f(r). We can then use
Newton's method to find the Gram point near a value of r where f(r)
is close to an integer G. In this case, if we ignore the terms in
inverse powers of r in the asymptotic formula we simply need iterate

r' = (G + r + 1/8) / ln(r)

If n is the number of steps in an n-et, then r = n/ln(2). For
example, if n=12 then r=17.32... and f(r) = 31.927..., so we pick the
Gram point where G=32. Applying Newton's method, we get an adjusted
value r=17.337... which corresponds to n=12.017..., where octaves are
flat by 1.764... cents. We may call such a tuning a Gram tuning, and
it is interesting to consider what the flatness or sharpness of the
Gram tuning for a given et is telling us, and whether they make
practical sense.

I found an amazing applet on the web, which will graph Z(t) for
anyone who wants to investigate this. (Don't you just love Java?)
Unfortunately, it isn't set up for microtonalists, but if you
multiply your et n by the magic number 2 pi / ln(2), you get a value
of t which corresponds to that et and around which it is interesting
to consider the graph. Maybe if we ask nicely he will put up a
version for music theorists; he might be interested by the interest!

The url is:

http://www.math.ubc.ca/~pugh/RiemannZeta/RiemannZetaLong.html