A231327 Denominator of rational component of zeta(4n)/zeta(2n).

1, 15, 105, 675675, 34459425, 16368226875, 218517792968475, 30951416768146875, 694097901592400930625, 23383376494609715287281703125, 2289686345687357378035370971875, 219012470258383844016431785453125, 4791965046290912124048163518904807546875

Offset: 0

Author

Leo Depuydt, Nov 07 2013

Keywords

• nonn
• frac

Comments

Denominator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} prime(k)^(2n)/(prime(k)^(2n)+1).

For a detailed account of the results in question, including proof and relation to the zeta function, see the PDF file submitted as supporting material in A231273.

The reference to Apostol below is a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{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). - Leo Depuydt, Nov 22 2013

Denominator of B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!) where B(n) are the Bernoulli numbers (see A027641 and A027642). - Robert Israel, Aug 22 2014

References

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

Crossrefs

Cf. A231273 (the corresponding numerator).

Cf. A114362 and A114363 (closely related results).

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

Cf. A027641 and A027642.

Programs

Maple

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

Mathematica

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