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

[jean-paul allouche]

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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