In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby counts each distinctprime factor, whereas the related function counts the total number of prime factors of honoring their multiplicity. For example, if we have a prime factorization of of the form for distinct primes , then the respective prime omega functions are given by and. These prime factor counting functions have many important number theoretic relations.
An average order of both and is. When is prime a lower bound on the value of the function is. Similarly, if is primorial then the function is as large as on average order. When is a power of 2, then Asymptotics for the summatory functions over,, and are respectively computed in Hardy and Wright as where is again the Mertens constant and the constant is defined by Other sums relating the two variants of the prime omega functions include and
In this example we suggest a variant of the summatory functions estimated in the above results for sufficiently large. We then prove an asymptotic formula for the growth of this modified summatory function derived from the asymptotic estimate of provided in the formulas in the main subsection of this article above. To be completely precise, let the odd-indexed summatory function be defined as where denotes Iverson's convention. Then we have that The proof of this result follows by first observing that and then applying the asymptotic result from Hardy and Wright for the summatory function over, denoted by, in the following form:
The computations expanded in Chapter 22.11 of Hardy and Wright provide asymptotic estimates for the summatory function by estimating the product of these two component omega functions as We can similarly calculate asymptotic formulas more generally for the related summatory functions over so-termed factorial moments of the function.
A known Dirichlet series involving and the Riemann zeta function is given by The function is completely additive, where is strongly additive. Now we can prove a short lemma in the following form which implies exact formulas for the expansions of the Dirichlet series over both and : Lemma. Suppose that is a strongly additive arithmetic function defined such that its values at prime powers is given by, i.e., for distinct primes and exponents. The Dirichlet series of is expanded by Proof. We can see that This implies that wherever the corresponding series and products are convergent. In the last equation, we have used the Euler product representation of the Riemann zeta function. The lemma implies that for, where is the prime zeta function and is the Liouville lambda function.
The distribution of the difference of prime omega functions
The distribution of the distinct integer values of the differences is regular in comparison with the semi-random properties of the component functions. For, let the sets These sets have a corresponding sequence of limiting densities such that for These densities are generated by the prime products With the absolute constant, the densities satisfy Compare to the definition of the prime products defined in the last section of in relation to the Erdős–Kac_theorem.