  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.
