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

Vladimir Reshetnikov v.reshetnikov at gmail.com
Fri Jan 2 15:56:24 CET 2009

```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.
What discussion group would you recommend for general math questions?

On Fri, Jan 2, 2009 at 3:46 PM, Olivier Gerard <olivier.gerard at gmail.com> wrote:
>
> This is simply not true.
>
> Just compute more precisely the first few cases. If you use Mathematica
> (as you seem to do) you will see for instance that the value at s=1/2 of the
> second derivative of Zeta is not an integer.
>
> Derivative[2][Zeta][1/2]
>
> -16.0083570139286614226913065059449627...
>
> or following your formulation
>
> (Derivative[2][Zeta][1/2]/2 + 2^(2 + 1) )
>
> is
>
> -0.00417850696433071134565325297248139...
>
> this is not zero and this is not a computation artefact.
>
>
> What you can try to prove instead, using classical analysis, is that
>
> d_(k+1) Zeta(1/2)   /   d_(k) Zeta(1/2)    converges to  2(k+1)  when k -> oo
>
> relatively quickly
>
> All (mathematics) related inquiries are welcome on the seqfan list, but
> I prefer that they be in direct relation to integer sequences.
>
> If there is no clear link to integer sequences or the content of the OEIS,
> consider posting this on other math oriented mailing lists or newsgroups.
>
> Also note that the art of conjecture requires a minimum of resistance from
> the conjectured property (that is: a minimum of force in the initial
> attack from the author
> of the conjecture) and at least an apparent void in the available means
> of proof.
>
>
> Olivier
>
>
> On Fri, Jan 2, 2009 at 14:47, Vladimir Reshetnikov
> <v.reshetnikov at gmail.com> wrote:
>> Hi
>>
>> Table[Round[-Derivative[k][Zeta][1/2]/k!] - 2^(k + 1), {k, 1, 50}]
>>
>> Conjecture: this sequence consists of all zeros.
>> Can anybody prove it?
>>
>> Thanks