The Dirichlet Series for the Liouville Function and the Riemann Hypothesis

TL;DR

This paper explores the properties of a Dirichlet series related to the Liouville function, using a novel summation method to demonstrate that the nontrivial zeros of the Riemann zeta function lie on the critical line, supporting the Riemann Hypothesis.

Contribution

It introduces a new summation technique for the Dirichlet series of the Liouville function and provides a proof that the zeros of the zeta function are on the critical line.

Findings

Demonstrates the Dirichlet series is analytic for Re(s) > 1/2 and Re(s) < 1/2

Shows a symmetry in prime factorization of integers affecting the series

Supports the Riemann Hypothesis by confirming zeros lie on the critical line

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.