030 031 number, we can derive the Zeta Function. Alas, another mouthful, let’s visualise this. This is the first of the infinite series that I mentioned. 2, which is the first prime number is the base. Then we are taking incremental powers of the prime number as we progress further in the series. Lastly, that is all raised to the power of s. This structure is followed for all prime numbers as shown here. I hope that you all now understand what the statement, “the summation of all one over the powers of a prime number raised to s”, means now. What’s the next step? The next step is to then multiply all these infinite series. As there are infinite prime numbers, it will be the infinite product of these infinite series; and that is the Zeta Function. The Riemann-Zeta Function The question to be asking now is, where have the prime numbers come from? How are they relevant to the Zeta Function at all? Just why? Well, if you were living in Euler’s time you would have been left on a terrible cliffhanger, because it took around a century before Bernhard Riemann began to delve deeply into the Zeta Function and its relation to the prime numbers. What was Riemann’s approach to the Zeta Function that led to it revolutionising it? Riemann was also one of the founding fathers of complex analysis. Complex analysis is a branch of mathematics that studies functions with complex inputs and outputs. Therefore, Riemann took a unique approach to the Zeta Function, quite literally a complex approach. Before, we had mentioned that the Zeta Function only converged when ‘s’ was strictly greater than 1. Riemann, using mathematics called analytic continuation, extended the Zeta Function beyond the limits of what Euler had discovered. This led to one of the most beautiful and sophisticated function that I have ever come across in my life. The Riemann-Zeta function. Unfortunately, I am not learned enough to explain to you all how Riemann’s ‘edition’ of the Zeta Function truly works and how this all relates back to prime number theory. However, I urge you all to read further about the Riemann-Zeta function and the Riemann Hypothesis. visualise this. If we let each infinite series be the summation of all ‘one over the powers of a prime number raised to s’ and then multiply that series by the same series for the next prime number, we can derive the zeta function. Alas, another mouthful, let’s visualise this. z ( ) = 2 1 % ( + 2 1 !" + 2 1 * ( + 2 1 +" … This is the first of the infinite seri s that I mentioned. 2, which is the first prime number is the base. Then we are taking incremental powers of the prime number as we progress further in the series. Lastly, that is all raised to the power of s. This structure is followed for all prime numbers as shown here. z ( ) = 2 1 % ( + 2 1 !" + 2 1 * ( + 2 1 +" … z ( ) = 3 1 ( + 3 1 !( + 3 1 *( + 4 1 +( … z ( ) = 5 1 ( + 5 1 !( + 5 1 *( + 5 1 +( … z ( ) = 7 1 ( + 7 1 !( + 7 1 *( + 7 1 +( … The Riemann-Zeta Function The question to be asking now is, where have the prime numbers come from? How are they relevant to the zeta function at all? Just why? Well, if you were living in Euler’s time you would have been left on a terrible clifhanger, because it took around a century before Bernhard Riemann began to delve deeply into the Zeta function and its relation to the prime numbers. What was Riemann’s approach to the zeta function that led to it revolutionising it? Riemann was also one of the founding fathers of complex analysis. Complex analysis is a branch of mathematics that studies functions with complex inputs and outputs. Therefore, Riemann took a unique approach to the zeta function, quite literally a complex approach. Before, we had mentioned that the zeta function only converged when ‘s’ was strictly greater than 1. Riemann, using mathematics called analytic co tinuati n, extend d the ze a function beyond the limits of what Euler had discovered. T is led to one of th most beau iful and sophisticated function that I have ever come across in my life. The Riemann-Zeta function. Unfortunately, I am not learned enough to explain to you all how Riemann’s ‘edition’ of the Zeta function truly works and how this all relates back to prime number theory. However, I urge you all to read further about the Riemann-Zeta function and the Riemann Hypothesis. If we let ach infinite series be the summation of all ‘one over the powers of a prime number raised to s’ and then multiply that series by the same series for the next prime number, we can derive the zeta function. Alas, another mouthful, let’s visualise this. z ( ) = 2 1 % ( + 2 1 !" + 2 1 * ( + 2 1 +" … This is the first of the infinite series that I mentioned. 2, which is the first prime number is the base. Then we are taking incremental powers of the prime number as we progress further in the series. Lastly, that is all raised to the power of s. This structure is followed for all prime numbers as shown here. z ( ) = 2 1 % ( + 2 1 !" + 2 1 * ( + 2 1 +" … z ( ) = 3 1 ( + 3 1 !( + 3 1 *( + 4 1 +( … z ( ) = 5 1 ( + 5 1 !( + 5 1 *( + 5 1 +( … z ( ) = 7 1 ( + 7 1 !( + 7 1 *( + 7 1 +( …
RkJQdWJsaXNoZXIy ODA2Njk=