https://riemannhypothesis.blog/tags/fundamental-theorem/ Recent content in Fundamental Theorem on Understanding the Riemann Hypothesis Hugo -- gohugo.io en markus@recommend.games (Markus Shepherd) markus@recommend.games (Markus Shepherd) Fri, 05 Aug 2016 15:38:11 +0000 https://riemannhypothesis.blog/2016/08/primes-from-a-different-world/ Fri, 05 Aug 2016 15:38:11 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2016/08/primes-from-a-different-world/ Every textbook on number theory will begin with a treatise on prime numbers; every treatise on prime numbers will begin by emphasising their importance as building blocks or atoms of our number system: every integer can be expressed as a product of prime numbers in one way and one way only. Six is two times three and there is no other way to decompose it.1 Euclid proved this over two thousand years ago and it is so fundamental (hence the name fundamental theorem of arithmetic) to our thinking about numbers that we take it for granted. https://riemannhypothesis.blog/2014/04/from-zeta-to-j-and-back-and-yet-again-back/ Sun, 06 Apr 2014 14:13:42 +0000markus@recommend.games (Markus Shepherd) https://riemannhypothesis.blog/2014/04/from-zeta-to-j-and-back-and-yet-again-back/ We know a lot about the \(\zeta\) and \(\xi\)-functions, we’ve learnt all about the different prime counting functions, most notably \(J(x)\), so it’s high time we found a connection between the two. Probably not too surprisingly, the crucial link is our good friend, the Euler product \[ \zeta(s)=\prod_{p}(1-p^{-s})^{-1}. \] What we want to develop now is a version of this product that will suit us to find a formula that magically can count primes.