Thursday, 10 March 2016

number theory - Does this show that the Apery Constant is transcendental?




Last August I posted this on mathoverflow: https://mathoverflow.net/questions/71856/a-serendipitous-riemann-identity. I show the (slightly revised) equation below:



ζ(3)=2π4315n=1(1(pn)2pn+1)



Since the constant, 2π4315 contains π, which is known to be transcendental, wouldn't this prove that ζ(3) is transcendental?



I have calculated the product through the first million primes and Mathematica's Element[product,Rationals] returns True. Also, I built a continued fraction of 18,500,045 elements.



The product converges to http://oeis.org/A082695




A paper that uses the product: http://jtnb.cedram.org/cedram-bin/article/JTNB_2004__16_1_107_0.pdf


Answer



No, because an infinite product of rationals is not necessarily rational.



For instance, n=1(114n2)=2π



is not rational.


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