[seqfan] Re: A strong estimate of w(2n) from below and above

[Tomasz Ordowski]

Hello everyone interested,

Note that f(s) = 1 / t(s) = zeta(s)^2 / zeta(2s)
= Sum_{n=1..oo} 2^omega(n) / n^s,
where omega(n) = A001221(n),
so w(s) = (1 - t(s)) / (1 + t(s))
= (f(s) - 1) / (f(s) + 1).

But what's next?

Thomas

śr., 23 lis 2022 o 12:27 Amiram Eldar <amiram.eldar at gmail.com> napisał(a):

> Hello,
>
> The sequence whose Dirichlet g.f. is zeta(2s)/zeta(s)^2 is A158522.
>
> Best,
> Amiram
>
>
> On Tue, Nov 22, 2022 at 7:53 PM jean-paul allouche <
> jean-paul.allouche at imj-prg.fr> wrote:
>
> > oops sorry, replace "2-21" with "D-21"
> > jp
> >
> > Le 22/11/2022 à 18:52, jean-paul allouche a écrit :
> > > Hi
> > >
> > > I have a (vague) suggestion: if s is > 1, then zeta(2s)/zeta(s)^2
> > > can be expressed as an "explicit" Dirichlet series namely sum(u(n)/n^s)
> > > where u(n) is some sequence depending on the number of prime factors
> > > of n --see, e.g., Formula 2.21 in H. W. Gould, Temba Shonhiwa, A
> catalog
> > > of interesting Dirichlet series, Missouri J. Math. Sci. 20, 2–-18
> (2008).
> > > Now replacing s by m, you get (hopefully) a series for t(m).
> > > Truncating this
> > > Dirichlet series might give you something like your conjecture.
> > >
> > > best wishes
> > > jean-paul
> > >
> > > Le 21/11/2022 à 18:51, Tomasz Ordowski a écrit :
> > >> Dear readers!
> > >>
> > >> Let w(m) = (1 - t(m)) / (1 + t(m)),
> > >> where t(m) = zeta(2m) / zeta(m)^2.
> > >>
> > >> For m = 2n, we have on the OEIS:
> > >> w(2n) = A348829(n) / A348830(n),
> > >> t(2n) = A114362(n) / A114363(n).
> > >>
> > >> Conjecture:
> > >> 0 < w(2n) - (1/2^{2n}+1/3^{2n}+1/5^{2n}+1/7^{2n}) < 1/11^{2n},
> > >> for every n > 0.
> > >>
> > >> Amiram Eldar confirmed my strong estimate up to n = 10^4.
> > >>
> > >> Is this conjecture provable?
> > >>
> > >> Best regards,
> > >>
> > >> Thomas Ordowski
> > >> _______________________
> > >> The strong estimate of w(2n) from below and above:
> > >> Sum_{prime p <= 7} 1/p^{2n} < w(n) < Sum_{prime p <= 11} 1/p^{2n},
> > >> for every n > 0.
> > >>
> > >> --
> > >> Seqfan Mailing list - http://list.seqfan.eu/
> > >
> > >
> > > --
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> >
> >
> > --
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> >
>
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