NHK_The_Cosmic_Code_Breakers.html [NHK] The Cosmic Code Breakers https://www.youtube.com/watch?v=eSOlPnnXnXA ================================================== All numbers can be represented by multiplication of prime numbers For example, $$$255=5\times 51$$$ $$$255=5\times (3\times 17)$$$ 5, 3, 17 can't be divided further So, numbers which can't be divided further are "prime numbers" ================================================== Prime numbers don't look showing up regularly For example, They show up quite in a row at some part But they don't show up for a while at some part ================================================== Riemann hypothesis: Non-trivial zero points of Zeta function will be on one line ================================================== Following equation which shows relationship between prime number and pi was created by Euler $$$\dfrac{2^2}{2^2-1} \times \dfrac{3^2}{3^2-1} \times \dfrac{5^2}{5^2-1} \times \dfrac{7^2}{7^2-1} \times \dfrac{11^2}{11^2-1} \times ... =\dfrac{\pi^2}{6}$$$ ================================================== Zeta function of Riemann $$$\zeta(x) = \prod\limits_p \dfrac{p^x}{p^x-1} = \dfrac{2^x}{2^x-1} \times \dfrac{3^x}{3^x-1} \times \dfrac{5^x}{5^x-1} \times \dfrac{7^x}{7^x-1} \times \dfrac{11^x}{11^x-1} \times ...$$$ ================================================== Graph of Zeta function ================================================== Above shows location of point (called zero point) which has zero height in the graph ================================================== Where zero point will show? ================================================== Riemann found 4 zero points And found zero points were on the one line like following ================================================== It's amazing because even if prime numbers show up in non-regular form, but zero points which are derived from zeta function and are more originally derived from prime numbers show up in regular form. ================================================== But since Riemann only found 4 zero points, he needed to think how about last all zero points? Would it be also on the one line like this? ================================================== Hypothesis of "Would it be also on the one line like this?" is that non-trivial zero points of Zeta function will be on one line ================================================== If above hypothesis is proved, it means that arrange of prime numbers has some regular rule in mathematical aspect ================================================== So, in conclusion, problem of finding rule in prime numbers is converted into problem of proving hypothesis saying all zero points are on the one line? ================================================== Hardy had proved that infinite zero points would be on the one line like this But he didn't remove probability that zero points could existing outside of that line ================================================== John Nash thought to prove Riemann hypothesis, he could need unique way of accessing to hypothesis ================================================== Dr. louis de branges says understanding prime numbers is related to dynamics in micro world ================================================== H. Montgomery found that unlike distribution of prime numbers which shows more irregular form of distribution zero points (in other words gap distance between each zero point) show more regular form of distribution ================================================== Distribution of zero points has above notation ================================================== Above notation is similar to notation which represents gap of energy level of nucleus of atom like Uranium ================================================== ================================================== Energy of nucleus of atom doesn't show in regular distribution ================================================== Relationship between distribution of zero points and energy level distribution of atom