[seqfan] Re: Euler product and prime zeta function

[Tomasz Ordowski]

   Maximilian,
I don't understand your objections, physical interpretation can be useful.
Reference to the physical model takes place, for example, in spatial
geometry.
Physicists create mathematical models of reality, and mathematicians can
use physics formulas, why not.

Coming back to my Conjecture
(which may be equivalent to the Riemann Hypothesis):
If a real number s > 0, then 0 < 1 - 1/zeta(s) - 1/2^s - 1/3^s < 1/5^s.

This Conjecture is equivalent to the statement:
If a real number s > 0, then Sum_{n>=5} mu(n)/n^s < 0 and Sum_{n>=6}
mu(n)/n^s > 0,
where mu(n) = A008683(n) is the Mobius function of n.

The complex numbers s for which these sums equal zero are respectively:
14.099850403365277... + i 0.37292303993962317... and
12.749725641163252... + i 1.2318590601928474...
[found by Amiram Eldar].

I end with wishes: happy New Year!

  Thomas


czw., 29 gru 2022 o 21:48 M. F. Hasler <oeis at hasler.fr> napisał(a):

> 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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> >
>
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