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Numerator of relativistic sum w(2n) of the velocities v = 1/p^(2n) over all primes p, in units where the speed of light c = 1.
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%I #95 Dec 21 2022 22:14:39

%S 3,1,12,59,521,872492,415603,471263387,100453109125251,

%T 249063001217323,1206701295264057,2340564635396243082668,

%U 1836709980831869650909,7917057291763619291770993,6790679763108188972468718224386027,497252110757159525928442098399943

%N Numerator of relativistic sum w(2n) of the velocities v = 1/p^(2n) over all primes p, in units where the speed of light c = 1.

%C Generally, for a complex number s, w(s) = tanh(Sum_{p prime} artanh(1/p^s)), assuming that Re(s) > 1.

%C 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.

%C 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.

%C If an integer s > 0 is even, then w(s) is rational (related to the Bernoulli numbers B_{s} and B_{2s}).

%C Conjecture: if an odd integer s > 1, then w(s) is irrational. Cf. W. Kohnen (link).

%C Note: Apery's constant zeta(3) = 1.202... is irrational.

%H Winfried Kohnen, <a href="https://doi.org/10.1007/BF02864395">Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms</a>, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, Vol. 99, No. 3 (1989), pp. 231-233.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Euler_product">Euler product</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Riemann_zeta_function">Riemann zeta function</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Velocity-addition_formula">Velocity-addition formula</a>.

%F a(n) = Numerator(tanh(Sum_{p prime} artanh(1/p^(2n))).

%F a(n) = Numerator((zeta(2n)^2-zeta(4n))/(zeta(2n)^2+zeta(4n))).

%F a(n) = Numerator((1-t(2n))/(1+t(2n))), where t(2n) = A114362(n)/A114363(n).

%F If Re(s) > 1, then w(s) = f(f(w(s))) = (1-t(s))/(1+t(s)) and t(s) = f(f(t(s))) = (1-w(s))/(1+w(s)) = zeta(2s)/zeta(s)^2, where f(x) = (1-x)/(1+x). See my theorem and the note under my proof of this theorem. - _Thomas Ordowski_, Jan 03 2022

%F 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 conjecture numerically up to n = 10^4. - _Thomas Ordowski_, Nov 13 2022

%e w(2) = 3/7, w(4) = 1/13, w(6) = 12/703, ...

%t r[s_] := Zeta[2*s]/Zeta[s]^2; w[s_] := (1 - r[s])/(1 + r[s]); Table[Numerator[w[2*n]], {n, 1, 15}] (* _Amiram Eldar_, Nov 01 2021 *)

%Y The denominators are A348830.

%Y Cf. A114362, A114363.

%Y See also A348131, A348132.

%K nonn,frac

%O 1,1

%A _Thomas Ordowski_, Nov 01 2021

%E More terms from _Amiram Eldar_, Nov 01 2021