# [seqfan] Re: Relativistic sum W(s) of the velocities 1/p^s over all primes p; ...

Tomasz Ordowski tomaszordowski at gmail.com
Tue Oct 12 13:40:24 CEST 2021

```P.S. As it was easy to see, if s > 0 is even, then
W(s) = (Zeta(s)^2 - Zeta(2s)) / (Zeta(s)^2 + Zeta(2s))
is rational (related to Bernoulli B_{s} and B_{2s}).
Conjecture: if odd s > 1, then W(s) is irrational.
Apery's constant Zeta(3) is irrational, see
Apéry's constant - Wikipedia
<https://en.wikipedia.org/wiki/Ap%C3%A9ry%27s_constant>

T. Ordowski

wt., 5 paź 2021 o 13:30 Tomasz Ordowski <tomaszordowski at gmail.com>
napisał(a):

> ... in Planck units, where the speed of light c = 1.
>
>
> Let us write the title sum as
> W(s) = tanh(Sum_{p prime} artanh(1/p^s)).
>
> Theorem: W(s) = (1-T(s)) / (1+T(s)),
> where T(s) = Zeta(2s) / Zeta(s)^2.
>
> For example: W(1) = 1, W(2) = 3/7, W(4) = 1/13.
> Let F(n) = N(n) / D(n) = W(2n) for n > 0. Data:
> 3/7, 1/13, 12/703, 59/14527, 521/524354, ...
>
> Proof of the Theorem.
> The formula w = (u+v)/(1+uv) can be extended ...
> We have (1-w)/(1+w) = (1-u)/(1+u) (1-v)/(1+v) ... Hence
> Product_{p prime} (1-1/p^s)/(1+1/p^s) = Zeta(2s)/Zeta(s)^2.
> Note that the function f(x) = (1-x)/(1+x) is an involution:
> Involution (mathematics) - Wikipedia
> <https://en.wikipedia.org/wiki/Involution_(mathematics)>
>
> Regards,
>
> T. Ordowski
> ___________
> Euler product - Wikipedia <https://en.wikipedia.org/wiki/Euler_product>
>
>

```