Sunday, August 26, 2018

[ohxflzck] The shape of the Riemann zeta function along the real line

The zeta function has a pole at s=1.  Proceeding rightward (positive-ward) from s=1, staying strictly along the real line, the value decreases from positive infinity asymptotically toward 1.  Some famous values are zeta(2)=pi^2/6=1.64, zeta(3)=Apery's Constant=1.20, zeta(4)=pi^4/90=1.08.  The function doesn't look anything special at those points.  The rate at which it approaches the asymptote seems to be negative exponential (geometric), as can be seen with Plot[Log[Zeta[x]-1], {x, 1, 20}].  What is the rate of decay?  This should be fairly easy because its values at even integers have closed form.

Proceeding leftward from the pole s=1, toward and along the negative real line, things are a little more exciting, but not as frequently reported.  The value starts at negative infinity then hits zero at s=-2.  Some famous points along the way are zeta(1/2, i.e., where it intersects the critical line)=-1.46, zeta(0)=-1/2=-0.5, zeta(-1)=-1/12=-0.083.  After hitting zero at s=-2, it oscillates around zero, hitting zero at all negative even numbers.  The local extrema between the zeroes are approximately at the negative odd integers, which gives an approximation of the amplitude of the oscillations: zeta(-3)=0.0083, zeta(-5)=-0.0040, zeta(-7)=0.0042, zeta(-9)=-0.0076, zeta(-11)=0.021, zeta(-13)=zeta(-1)=-0.083, zeta(-15)=0.44, zeta(-17)=-3.1, zeta(-19)=26., zeta(-21)=-280.  Incidentally, the values at these negative odd integers are exact rational numbers related to Bernoulli numbers.  After reaching a minimum around -5 or -7, the amplitudes grow faster than exponentially to the left, which is a little bit surprising after staying small for so long.  The growth rate is something like O(n^n) due to the fast rate of growth of the Bernoulli numbers: Plot[Log[Abs[Zeta[x]]], {x, -40, 4}]

Here are some aesthetically nice regions.  This is to generate nice irrationals, after discovering that MPFR offers the zeta function.

zeta(1.10621) = 10
zeta(3.77062) = 1.1
zeta(6.75097) = 1.01

Outside the above range, values get aesthetically too large or too close to 1.

zeta(0.905395) = -10
zeta(-18.4757) = 10

Outside the above range, values get too large.  Perhaps omit from -2 to -14, -16, or -18, to avoid a region of somewhat small values (but they are interesting).

Domain limits were calculated with Mathematica FindRoot.

The zeta function is analytic, so everything interesting about its behavior over the entire complex plane (including the million-dollar critical strip) is theoretically encoded in its behavior on the real line.