On The Hidden Structure of Odd Numbers and Its Consequences for the Riemann Hypothesis

Abstract

The Riemann hypothesis remains unconfirmed or invalidated to this day, although local verifications on the calculation of its zeros have never found it faulty. Mertens reformulated the problem to make it much more accessible and surely more easily solvable. Unfortunately, his conjecture, also called a strong conjecture, turned out to be incorrect. There remain 2 other conjectures, the weak and the general, which do not yet have fixed status. Would it then be possible that the resolution of the Riemann hypothesis arises through one of these 2 conjectures? We answer yes, and we turn our attention to Mertens' weak conjecture. Equipped with a new equation to date and a methodical approach that uses a bounded description of numbers, we solve the conjecture by placing ourselves under the criteria of Hausdorff's theorem concerning the evolution of the sum, by showing first of all that odd numbers have a structure similar to that of triangular numbers, and then the randomness arises from their intrinsic regularity; which does not contradict the Martin-L\"{o}f definition of random sequences despite everything. We therefore resolve the Riemann hypothesis and we provide an equation which will certainly make it possible to resolve other types of problems, and thus to extend the means made available to mathematicians to examine various types of questions whether in number theory or in other fields of mathematics, or even in physics, cryptography and computer science.