Assigning of values of the prime number counting function to Bernhard Riemann's zeros. Concept of Dirichlet lines in the complex plane

Abstract or Introduction

Given their importance as the building blocks of all of the integers, the prime numbers, their number and dispositionwithin the integers have been studied for millennia. In 1859 Bernhard Riemann published a short paper (which only extended to six manuscript pages) concerned with an investigation of the number of prime numbers less than any given number. The outcome of the work revolutionised number theory and in the years following has resulted in almost what could be termed an industry in a particular aspect of his work, now called the Riemann Hypothesis. Riemann showed that all the zero values of the zeta function in the positive part of the complex plane should lie in a region between x = 0 and x = 1, and, in particular, conjectured that they all should lie on the line of symmetry x = ½.

This conjecture, which is the basis of many other conjectures in Number Theory is considered by many mathematicians to be the currently greatest unsolved problem in Mathematics—so much so that the Clay Mathematical Institute of Boston, Mass. has offered one million dollars to anyone who can produce a solution. No attempt is made here to verify the hypothesis, for its validity is not required. A procedure is developed here, by means of which, the zeros of Riemann's zeta function in the so-called Critical Strip of the complex plane may be assigned values of the prime number counting function. The procedure is novel and uses the concept of a Dirichlet line in the complex plane and a quantity called a nearodd (both of which are defined in the text).

The process may be rendered self-contained, in the sense that, when it is associated with Gram's series the only input required is the magnitude of the imaginary part of the function s = x + iy which will locate a Riemann zero on the Critical Line; vast numbers of zeros may be accessed in [2]. Further, it is shown that the Riemann conjecture is irrelevant in the assigning of any particular value of the prime number counting function to the corresponding Riemann zero. It is suggested, pace Wiles, who obtained a proof of Fermat's Last Theorem as a by-product of his verification of the Taniyama-Shimura conjecture, that, in the light of Godel's incompleteness theorems, Riemann's hypothesis may be undecidable.