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

[Amiram Eldar]

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.
> >>
> >> --
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> >
> >
> > --
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>
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