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A231273 Numerator of zeta(4n)/(zeta(2n) * Pi^(2n)). 2
1, 1, 1, 691, 3617, 174611, 236364091, 3392780147, 7709321041217, 26315271553053477373, 261082718496449122051, 2530297234481911294093, 5609403368997817686249127547, 61628132164268458257532691681, 354198989901889536240773677094747 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Integer component of the numerator of a close variant of Euler's infinite prime product zeta(2n) = prod_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus, as follows: zeta(4n)/zeta(2n) = prod_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)+1). The transcendental component is Pi^(2n).

For a detailed account of the results, including proof and relation to the zeta function, see Links for the PDF file submitted as supporting material.

The reference to Apostol is to a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, prod_{sigma>1} (prime(n)^(s)/(prime(n)^(s) + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). On this, see also Hardy and Wright cited below. - Leo Depuydt, Nov 22 2013, Nov 27 2013

The background of the sequence is now described in the link below to L. Depuydt, The Prime Sequence ... . - Leo Depuydt, Aug 22 2014

From Robert Israel, Aug 22 2014: (Start)

Numerator of (-1)^n*B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!), where B(n) are the Bernoulli numbers (see A027641 and A027642).

Not the same as abs(A001067(2*n)): they differ first at n=17.

(End)

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, p. 231.

Leo Depuydt, The Prime Sequence: Demonstrably Highly Organized While Also Opaque and Incomputable-With Remarks on Riemann's Hypothesis, Partition, Goldbach's Conjecture, ..., Advances in Pure Mathematics, 4 (No. 8, 2014), 400-466.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, 1960, p. 255.

LINKS

Table of n, a(n) for n=0..14.

Leo Depuydt, Details, including proof and relation to zeta function

MAPLE

seq(numer((-1)^n*bernoulli(4*n)*4^n*(2*n)!/(bernoulli(2*n)*(4*n)!)), n=0..100); # Robert Israel, Aug 22 2014

MATHEMATICA

Numerator[Table[Zeta[4n]/(Zeta[2n] * Pi^(2n)), {n, 0, 15}]] (* T. D. Noe, Nov 18 2013 *)

CROSSREFS

Cf. A231327 (corresponding denominator).

Cf. A114362 and A114363 (closely related results).

Cf. A001067, A046968, A046988, A098087, A141590, A156036 (same number sequence, though in various transformations (alternation of signs, intervening numbers, and so on)).

Cf. A027641 and A027642

Sequence in context: A327448 A046753 A033563 * A156036 A029814 A135843

Adjacent sequences:  A231270 A231271 A231272 * A231274 A231275 A231276

KEYWORD

nonn,frac

AUTHOR

Leo Depuydt, Nov 07 2013

STATUS

approved

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Last modified December 8 09:32 EST 2019. Contains 329862 sequences. (Running on oeis4.)