[seqfan] Prime zeta and Riemann zeta functions
Tomasz Ordowski
tomaszordowski at gmail.com
Thu Nov 17 15:35:26 CET 2022
Dear readers!
Let P(n) = Sum_{prime p} 1/p^n, for n > 1.
Theorem:
P(n) ~ w(n) with w(n) < P(n) for n > 1,
where w(n) = (1 - t(n)) / (1 + t(n))
and t(n) = zeta(2n) / zeta(n)^2.
Conjecture:
w(n) = 1/2^n + 1/3^n + 1/5^n + 1/7^n + O(1/11^n),
where t(n) and w(n) as above (and below)*.
Is this conjecture provable?
Best regards,
Thomas Ordowski
_________________
(*) For even numbers, we have
t(2n) = A114362(n) / A114363(n),
w(2n) = A348829(n) / A348830(n).
On the OEIS pages:
A114362 - OEIS <https://oeis.org/A114362> / A114363 - OEIS
<https://oeis.org/A114363>
A348829 - OEIS <https://oeis.org/A348829> / A348830 - OEIS
<https://oeis.org/A348830>
where I have submitted my drafts ...
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