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

jean-paul allouche jean-paul.allouche at imj-prg.fr
Tue Nov 22 18:53:16 CET 2022

```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 :
>>
>> 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/
>
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/

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