Polynomial on Understanding the Riemann Hypothesis
https://riemannhypothesis.info/tags/polynomial/
Recent content in Polynomial on Understanding the Riemann HypothesisHugo -- gohugo.iomarkus@recommend.games (Markus Shepherd)markus@recommend.games (Markus Shepherd)Sun, 01 Dec 2013 21:29:17 +0000More symmetry and Another Product
https://riemannhypothesis.info/2013/12/more-symmetry-and-another-product/
Sun, 01 Dec 2013 21:29:17 +0000markus@recommend.games (Markus Shepherd)https://riemannhypothesis.info/2013/12/more-symmetry-and-another-product/We’ve seen that \(\zeta(s)\) satisfies the functional equation
\[ \zeta(1-s)=2^{1-s}\pi^{-s}\cos(\pi s/2)\Pi(s-1)\zeta(s). \]
(Well, it still needs to be proved, but let’s just assume it’s correct for now.) The goal of this post is an even more symmetrical form that will yield the function \(\xi(s)\) which we can develop into an incredibly useful product expression.
On our wish list for \(\xi(s)\) we find three items:
It’s an entire function, i.e., a function that’s holomorphic everywhere in \(\mathbb{C}\) without any poles.