[seqfan] Relativistic sum W(s) of the velocities 1/p^s over all primes p; ...
Tomasz Ordowski
tomaszordowski at gmail.com
Tue Oct 5 13:30:43 CEST 2021
... in Planck units, where the speed of light c = 1.
Dear readers!
Let us write the title sum as
W(s) = tanh(Sum_{p prime} artanh(1/p^s)).
Theorem: W(s) = (1-T(s)) / (1+T(s)),
where T(s) = Zeta(2s) / Zeta(s)^2.
For example: W(1) = 1, W(2) = 3/7, W(4) = 1/13.
Let F(n) = N(n) / D(n) = W(2n) for n > 0. Data:
3/7, 1/13, 12/703, 59/14527, 521/524354, ...
Proof of the Theorem.
The formula w = (u+v)/(1+uv) can be extended ...
We have (1-w)/(1+w) = (1-u)/(1+u) (1-v)/(1+v) ... Hence
Product_{p prime} (1-1/p^s)/(1+1/p^s) = Zeta(2s)/Zeta(s)^2.
Note that the function f(x) = (1-x)/(1+x) is an involution:
Involution (mathematics) - Wikipedia
<https://en.wikipedia.org/wiki/Involution_(mathematics)>
Regards,
T. Ordowski
___________
Euler product - Wikipedia <https://en.wikipedia.org/wiki/Euler_product>
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