# [seqfan] Re: n-th derivative of Zeta function

Olivier Gerard olivier.gerard at gmail.com
Fri Jan 2 16:59:59 CET 2009

```On Fri, Jan 2, 2009 at 15:56, Vladimir Reshetnikov
<v.reshetnikov at gmail.com> wrote:
> Dear Oliver,
>
> Thanks for your explanation. Please note the presence of Round (i.e.
> nearest integer function) in the formula - this makes equality to 0
> exact.

Indeed. I did not take that into account while reading your
Mathematica expression
by sight. So I misrepresented your claim. My sincere apologies.

The problem with Round/Floor/Ceiling is that they may be difficult to
deal with in formulas because of their discountinuities.

One could then consider several derived integer sequence such as the
sign of the difference between the Zeta derivative value and the
rational.

Note that the ratio between successive
derivatives of Zeta at integer and rational points converge to rationals
and that these properties follow symmetries of Zeta.

For instance

d_(k+1) Zeta(-1/2)   /   d_(k) Zeta(-1/2)    converges to  2/3 (k+1)
when k -> oo

d_(k+1) Zeta(5/2)   /   d_(k) Zeta(5/2)    converges to  2/3 (k+1)  when k -> oo

d_(k+1) Zeta(-3/2)   /   d_(k) Zeta(-3/2)    converges to  2/5 (k+1)
when k -> oo

d_(k+1) Zeta(7/2)   /   d_(k) Zeta(7/2)    converges to  2/5 (k+1)  when k -> oo

d_(k+1) Zeta(2/3)   /   d_(k) Zeta(2/3)    converges to  3 (k+1)  when k -> oo

d_(k+1) Zeta(4/3)   /   d_(k) Zeta(4/3)    converges to  3 (k+1)  when k -> oo

etc.

> What discussion group would you recommend for general math questions?
>

All have their qualities and defaults and I don't know all of them.

sci.math  is widely read has a lot of noise but helpful contributors.
math-fun is populated by interesting fellows.
I appreciate NMBRTHRY and FOM but they are more specialized.

General and unmoderated forums tend to be attacked by cranks.

Some mathematician blogs have become very valuable resources.

May be other seqfan members can suggest other forums of interest.

Regards,

Olivier

```