On a pair of zeta functions.

Let be a positive integer, and define for , where denotes the number of distinct prime factors of , and represents the total number of prime factors of (counted with multiplicity). In this paper, we study these two zeta functions and related arithmetical functions. We show that which is similar to the known identity equivalent to the Prime Number Theorem. For , we prove that We also raise a hypothesis on the parities of which implies the Riemann Hypothesis. [ABSTRACT FROM AUTHOR]