The Dirichlet Series for the Liouville Function and the Riemann Hypothesis

Abstract

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a novel method of summing the series by casting it as an infinite number of sums over sub-series that exhibit a certain symmetry and rapid convergence. In this procedure, which heavily invokes the prime factorization theorem, each sub-series has the property that it oscillates in a predictable fashion, rendering the analytic properties of the Dirichlet series determinable. With this method, the paper demonstrates that, for every integer with an even number of primes in its factorization, there is another integer that has an odd number of primes (multiplicity counted) in its factorization. Furthermore, by showing that a sufficient condition derived by Littlewood (1912) is satisfied, the paper demonstrates that the function F(s) is analytic over the two half-planes Re(s) > 1/2 and Re(s)<1/2. This establishes that the nontrivial zeros of the Riemann zeta function can only occur on the critical line Re(s)=1/2.