A348830 -id:A348830 - cp's OEIS frontend

A114362 Numerator of zeta(4n)/zeta(2n)^2 (with a(0)=2 instead of -2).

2, 2, 6, 691, 7234, 523833, 3545461365, 3392780147, 15418642082434, 26315271553053477373, 261082718496449122051, 2530297234481911294093, 39265823582984723803743892829, 61628132164268458257532691681

Offset: 0

Benoit Cloitre, Feb 09 2006; corrected Feb 22 2006

zeta(4n)/zeta(2n)^2 is a rational value expressible in term of Bernoulli's numbers (A027641).
Conjecture: if an integer n > 1 is odd, then zeta(2n)/zeta(n)^2 is irrational. Cf. W. Kohnen (link) and my conjecture in A348829. - Thomas Ordowski, Jan 05 2022
Conjecture: (1 - t(n))/(1 + t(n)) = 1/2^n + 1/3^n + 1/5^n + 1/7^n + O(1/11^n), where t(n) = zeta(2n)/zeta(n)^2. Cf. A348829. - Thomas Ordowski, Nov 13 2022

			2/1, 2/5, 6/7, 691/715, 7234/7293, 523833/524875, 3545461365/3547206349, ...

Seiichi Manyama, Table of n, a(n) for n = 0..158
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.
Herbert Wilf, Problem 11068, The American Mathematical Monthly, Vol. 111, No. 3 (2004), p. 259; Think Rationally, Solution to Problem 11068 by Kenneth E. Schilling, ibid., Vol. 112, No. 9 (2005), pp. 844-845.

Cf. A000984, A027641, A027642, A114363 (denominators), A348829, A348830.

a[n_] := Numerator[Zeta[4*n]/Zeta[2*n]^2]; a[0] = 2; Array[a, 14, 0] (* Amiram Eldar, Mar 04 2023 *)

z(n)=bernfrac(2*n)*(-1)^(n - 1)*2^(2*n-1)/(2*n)!;
a(n)=if(n<1,2,numerator(z(2*n)/z(n)^2))

Product_{p primes} (p^{2n}-1)/(p^{2n}+1) = zeta(4n)/zeta(2n)^2.
For n > 0, a(n) = Numerator((D(n) - N(n)) / (D(n) + N(n))), where N(n) = A348829(n) and D(n) = A348830(n). See my comments and formulas in A348829. - Thomas Ordowski, Jan 05 2022
From Amiram Eldar, Mar 04 2023: (Start)
a(n)/A114363(n) = -2*B(4*n)/(binomial(4*n,*2n)*B(2*n)) = -2*(A027641(4*n)/A027642(4*n))/(A000984(2*n)*A027641(2*n)/A027642(2*n)), for n >= 1, where B(n) is the n-th Bernoulli number.
A114363(n)/a(n) = Sum_{x in Q+} 1/f(x)^(2*n), for n >= 1, where Q+ is the set of the positive rational numbers, and if x = k/m in lowest terms, then f(x) = k*m (Wilf, 2004). (End)

A114363 Denominator of zeta(4n)/zeta(2n)^2.

Original entry on oeis.org

1, 5, 7, 715, 7293, 524875, 3547206349, 3393195750, 15419113345821, 26315472459271727875, 261083216622451556697, 2530298441183206558150, 39265828264113994596230058165, 61628134000978439089402342590

Offset: 0

Benoit Cloitre, Feb 09 2006; corrected Feb 22 2006

zeta(4n)/zeta(2n)^2 is a rational value expressible in term of Bernoulli's numbers (A027641).

			-2/1, 2/5, 6/7, 691/715, 7234/7293, 523833/524875, 3545461365/3547206349, ...

Seiichi Manyama, Table of n, a(n) for n = 0..158

Cf. A027641, A027642, A114362 (numerators), A348829, A348830.

a[ n_] := If[ n < 0, 0, Denominator[ Zeta[4*n] / Zeta[2*n]^2 ]] (* Michael Somos, Jan 27 2012 *)

z(n)=bernfrac(2*n)*(-1)^(n - 1)*2^(2*n-1)/(2*n)!;
a(n)=if(n<1,1,denominator(z(2*n)/z(n)^2))

For n > 0, Product_{p primes} (p^{2n}-1)/(p^{2n}+1) = zeta(4n)/zeta(2n)^2.

For n > 0, a(n) = Denominator((D(n) - N(n)) / (D(n) + N(n))), where N(n) = A348829(n) and D(n) = A348830(n). See my comments and formulas in A348829. - Thomas Ordowski, Feb 12 2022

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.

Original entry on oeis.org

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

Offset: 1

Thomas Ordowski, Nov 01 2021

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.

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

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.

The denominators are A348830.
Cf. A114362, A114363.
See also A348131, A348132.

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 *)

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

More terms from Amiram Eldar, Nov 01 2021

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A114362 Numerator of zeta(4n)/zeta(2n)^2 (with a(0)=2 instead of -2).

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A114363 Denominator of zeta(4n)/zeta(2n)^2.

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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.

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