[seqfan] A348829 / A348830
Tomasz Ordowski
tomaszordowski at gmail.com
Tue Nov 2 08:48:20 CET 2021
T. Ordowski
Relativistic sum w(s) of the velocities 1/p^s over all primes p,
in natural physical units where the speed of light c = 1.
GENERALLY, for a complex number s,
w(s) = tanh(Sum_{p prime} artanh(1/p^s)),
assuming that Re(s) > 1.
THEOREM.
If Re(s) > 1,
then w(s) = (1 - t(s)) / (1 + t(s))
with t(s) = zeta(2s) / zeta(s)^2,
where zeta(z) is the Riemann zeta function of z.
PROOF.
Einstein's formula w = (u + v)/(1 + uv) can be expanded as
(1-w)/(1+w) = ((1-u)/(1+u))((1-v)/(1+v))...
for any number of velocities u, v, ...
Hence, by the Euler product,
Product_{p prime} (1-1/p^s)/(1+1/p^s) =
= zeta(2s)/zeta(s)^2, QED.
Note that the function f(x) = (1-x)/(1+x) is an involution.
If an integer s > 0 is even, then w(s) is rational
(related to the Bernoulli numbers B_{s} and B_{2s}).
For example w(2) = 3/7, w(4) = 1/13, ... [*]
CONJECTURE.
If an odd integer s > 1, then w(s) is irrational.
Note: Apery's constant zeta(3) = 1.202... is irrational.
For example w(3) = 0.1736589933..., w(5) = 0.03574997731...
___________________
[*] w(2n) = A348829(n) / A348830(n).
See the comments and formulas sections.
The OEIS; Thomas Ordowski, Nov 01 2021
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