RIEMANN HYPOTHESIS REFUTATION

In this study, a set of new numbers, infinite - thalas numbers, is introduced. Infinite numbers are defined as limits of complex functions that tend to infinity. In infinity, two arguments are added; the velocity at which a complex function tends to infinity and its angle on the complex plane. This way a quantification of infinity is obtained and new numbers, the infinite numbers are determined. Firstly, the infinity unit is defined which is the limit of a function tending to infinity with i) a velocity equal to the real unit and ii) a zero angle on the complex plane. Based on the infinity unit definition, all infinite numbers are also defined. The logarithm of the infinity unit is obtained which is called log-infinity unit, and furthermore the log-infinite numbers are also determined. Considering an ortho-normal axle system, with units on the three vertical axes, respectively, the real numbers unit, the imaginary unit (i), and the log-infinity unit, a broader set of new numbers in the three-dimensional space is introduced, named thalas numbers. These new numbers are proved to be useful with interesting properties.

On the basis of infinite numbers and infinite geometric triangles, having sides that are infinite numbers, the well-known unsolved, mathematical problem “Riemann hypothesis” was investigated and solved. The solution showed that there are no zeros of the Riemann zeta function ζ (s) where s = α + ib, in the range (0, 1) of the real variable α. Therefore, even the known “zeros”, calculated by numerical methods, on the critical line of the complex plane (0.5 + ib) where b is a real number, in fact they are not zeros but are points where the zeta function takes values very close to zero. The solution-refutation of the “Riemann hypothesis” is presented in Part II of this paper, Chapters 17 - 21. To solve this problem, from the Part I concerning the presentation of infinite numbers, only Chapters 1, 2, 3, 4, 10, 11 and 13, are required. The proof of Riemann hypothesis is elegant, without long, complicated, obscure and questionable calculations.