[seqfan] A348829 / A348830

[Tomasz Ordowski]

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