[seqfan] Re: Euler product and prime zeta function

[M. F. Hasler]

Also note that appearance of the function (1-z)/(1+z) does not justify in
any way to speak of relativistic addition of velocities.

It's as if you would call any relation of proportionality, of the form y =
lambda x,
Newton's law or Hooke's law or Ohm's law, just because these laws are also
or that form, F = m a, F = k s, U = R I.
Unless the z in that formula does have some connection to some velocity
(displacement of what from where to where?), it is highly unscientific and
nonsensical to allude to relativistic addition of velocities, and such
non-sense should be banned from a scientific reference work as this
encyclopedia.

- Maximilian

On Mon, Dec 26, 2022, 23:37 Tomasz Ordowski <tomaszordowski at gmail.com>
wrote:

> P.S. Note that
> P(n)  >~ 1 - 1/zeta(n),
> where P(n) = Sum_{prime p} 1/p^n.
>
> Conjecture:
> 0 < 1 - 1/zeta(n) - 1/2^n - 1/3^n < 1/5^n,
> for every n > 1.
>
> T. Ordowski
>
> śr., 14 gru 2022 o 17:43 Tomasz Ordowski <tomaszordowski at gmail.com>
> napisał(a):
>
> > Dear readers!
> >
> > By the Euler product,
> > Product_{prime p} (1-1/(2p^n-1))/(1+1/(2p^n-1)) =
> > = Product_{prime p} (1-1/p^n) = 1/zeta(n), for n > 1.
> >
> > Note that w(n) = (1-1/zeta(n)/(1+1/zeta(n) = (zeta(n)-1)/(zeta(n)+1)
> > is the relativistic sum of the velocities v = 1/(2p^n-1) over all primes
> > p,
> > in units where the speed of light c = 1. Cf. A348829 / A348830.
> >
> > So, according to the above physical interpretation,
> > 1/(2*2^n-1) < w(n) = (zeta(n)-1)/(zeta(n)+1) < Sum_{prime p} 1/(2p^n-1).
> > The prime zeta P(n) = Sum_{prime p} 1/p^n ~ 2(zeta(n)-1)/(zeta(n)+1),
> > (but P(n) < 2w(n) for n > 1). Hence P(n) ~ log(zeta(n)) ~ zeta(n)-1.
> >
> > Let a(n) be the smallest prime q such that
> > Sum_{prime p <= q} 1/(2p^n-1) > (zeta(n)-1)/(zeta(n)+1).
> >
> > Is a(n) = Q for every n >= N (until the end of the calculation)?
> > If so, what are the hypothetical values of Q and N ?
> >
> > Best regards,
> >
> > Thomas Ordowski
> > __________________
> > Cf. A348829 / A348830.
> > See my draft with a new conjecture in the formula section:
> > The On-Line Encyclopedia of Integer Sequences® (OEIS®)
> > <https://oeis.org/history/view?seq=A348829&v=94>
> >
> >
>
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