The OEIS mourns the passing of Jim Simons and is grateful to the Simons Foundation for its support of research in many branches of science, including the OEIS.
The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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
 3, 1, 12, 59, 521, 872492, 415603, 471263387, 100453109125251, 249063001217323, 1206701295264057, 2340564635396243082668, 1836709980831869650909, 7917057291763619291770993, 6790679763108188972468718224386027, 497252110757159525928442098399943 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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}). 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} artanh(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 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. See also A348131, A348132. Sequence in context: A210587 A019232 A185697 * A321697 A263008 A016479 Adjacent sequences: A348826 A348827 A348828 * A348830 A348831 A348832 KEYWORD nonn,frac AUTHOR Thomas Ordowski, Nov 01 2021 EXTENSIONS More terms from Amiram Eldar, Nov 01 2021 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 27 02:37 EDT 2024. Contains 372847 sequences. (Running on oeis4.)