A348829 - OEIS

A348829

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.

3, 1, 12, 59, 521, 872492, 415603, 471263387, 100453109125251, 249063001217323, 1206701295264057, 2340564635396243082668, 1836709980831869650909, 7917057291763619291770993, 6790679763108188972468718224386027, 497252110757159525928442098399943

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OFFSET

1,1

COMMENTS

Generally, for a complex number s, w(s) = tanh(Sum_{p prime} arctanh(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}).

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

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

LINKS

Table of n, a(n) for n=1..16.

Winfried Kohnen, Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, Vol. 99, No. 3 (1989), pp. 231-233.

Wikipedia, Euler product.

Wikipedia, Riemann zeta function.

Wikipedia, Velocity-addition formula.

FORMULA

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

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

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

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

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

It can be proven that P(2n) - w(2n) ~ 1/12^(2n), where P(x) = Sum_{prime p} 1/p^x = 1/2^x + 1/3^x + 1/5^x + ... is the prime zeta function of real x > 1. - Thomas Ordowski, Nov 06 2024

EXAMPLE

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

MATHEMATICA

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 *)

CROSSREFS

The denominators are A348830.

Cf. A114362, A114363.