Authors: Pedro Caceres
PrimeNumbers are the atoms of mathematics and mathematics is needed to make sense of the real world. Finding the Prime number structure and eventually being able to crack their code is the ultimate goal in what is called Number Theory. From the evolution of species to cryptography, Nature finds help in Prime numbers. One of the most important advance in the study of Prime numbers was the paper by Bernhard Riemann in November 1859 called “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” (On the number of primes less than a given quantity). In this paper, Riemann gave a formula for the number of primes less than x in terms the integral of 1/log(x) and the roots (zeros) of the zeta function, defined by: (1)ζ(z) = ∑1/n^z The Zeta function, ζ(z), is a function of a complex variable z that analytically continues the Dirichlet series. Riemann also formulated a conjecture about the location of these zeros, which fall into two classes: the "trivial zeros" -2, -4, -6, etc., and those whose real part lies between 0 and 1. Riemann's conjecture Riemann hypothesis [RH] was formulated as this: [RH]The real part of every non-trivial zero z* of the Riemann Zeta function is 1/2. Thus, if the hypothesis is correct, all the non-trivial zeros lie on the critical line consisting of the complex numbers 1/2 + i ß, where ß is a real number and i is the imaginary unit. In this paper, we will analyze the Riemann Zeta function and provide an analytical/geometrical proof of the Riemann Hypothesis. The proof will be based on the fact that if we decompose the ζ(z) in a difference of two functions, both functions need to be equal when ζ(z)=0, so their distance to the origin or modulus must be equal and we will prove that this can only happen when Re(z)=1/2 for certain values of Im(z). We will also prove that all non-trivial zeros of ζ(z) in the form z=1/2+iß have all ß related by an algebraic expression. They are all connected and not independent. Finally, we will show that as a consequence of this connection of all ß, the harmonic function Hn can be expressed as a function of each ß zero of ζ(z) with infinite representations. We will use mathematical and computational methods available for engineers.
Comments: 25 Pages.
[v1] 2017-03-11 10:45:33
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