Derivatives of the zeta function on the critical line

The phenomenon of the peak values of zeta(1/2+it) being dominated by
near integers n = ln(2)t/(2 pi) turns out to be even more apparent
when you look at derivatives. While the real and imaginary parts still
oscillate like mad, combining them in the absolute value smooths
things out. The result is that the equal temperament peaks are more
apparent. The second derivative is even more smoothed out, and the
third looks a lot like the second.

Doing this also gives higher weight to the larger primes, and moves
the peak values for some tunings even farther away from the integers,
to adjusted flat or sharp tunings. I've put plots of the absolute
value of the derivative and second derivative up to 45 equal in the
Photos section; I also added a plot for 270, which shows that for the
later, really strong divisions the peaks start to look downright
alpine, and for 63-65, so you can contemplate how a confused division
like 64 plots out (which turns out to be in a confused way.)

Why no one notices this stuff I don't know. I've decided to try find
something interesting to prove, despite not being an analyst.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@c...> wrote:
>
> The phenomenon of the peak values of zeta(1/2+it) being dominated by
> near integers n = ln(2)t/(2 pi) turns out to be even more apparent
> when you look at derivatives.

Here's an explanation of the relationship between looking at zeta
peaks and the derivative. We have

zeta(1/2+it) = exp(i theta(t))Z(t)

where Z is the Z function and theta is the Riemann Siegel theta
function. I've recently written Wikipedia articles on these

http://en.wikipedia.org/wiki/Z_function

http://en.wikipedia.org/wiki/Riemann_Siegel_theta_function

From the above, we have

zeta'(1/2+it) = exp(I theta(t)(theta'(t)Z(t) - i Z'(t))

If we are at an extreme value for Z(t), so that Z'(t)=0, this reduces to

zeta'(1/2+it) = theta'(t) exp(I theta(t))Z(t) = theta'(t) zeta(1/2+it)

Hence, at the extreme points of Z, we have

|zeta(1/2+it)| = |zeta'(1/2+it)/theta'(t)|

so we are getting the same value, times a multiplier. From

theta(t) ~ (t/2)ln(t/(2pi))-t/2-pi/8+1/(48t)+...

we get

theta'(t) ~ (1/2)ln(t/(2pi)) + 1/(48t^2) + ...

and we are basically multiplying by a log factor. We can divide out by
theta' if we like, and the two systems become very comparable, but the
derivative gets rid of the rapid oscillation.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@c...> wrote:

> |zeta(1/2+it)| = |zeta'(1/2+it)/theta'(t)|
>
> so we are getting the same value, times a multiplier. From

I've uploaded a plot of |zeta| in green and |zeta'|/theta in red to
the zeta derivatives folder, under zeta, in Photos. It's quite
illuminating.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@c...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@c...> wrote:
>
> > |zeta(1/2+it)| = |zeta'(1/2+it)/theta'(t)|
> >
> > so we are getting the same value, times a multiplier. From
>
> I've uploaded a plot of |zeta| in green and |zeta'|/theta in red to
> the zeta derivatives folder, under zeta, in Photos. It's quite
> illuminating.

This is what first got me interested in the tuning-math group.
Unfortunately, in spite of reading 2 popular books on the RZH
("Prime Obsession" and "The Riemann Hypothesis") and looking at two
books for mathematicians on the subject, there is still a lot I
don't understand about it. Do you suppose, Gene, that you could add
a section to your webpage on RZF and RZH? Bringing us all up to
speed on how they relate to tuning-math considerations (I know...
you have plenty of old posts on the subject, but they are hard-to-
find this much later...)

Paul