In number theory, the prime omega functions
(little omega) counts each distinct prime factor, whereas the related function
(big omega) counts the total number of prime factors of
honoring their multiplicity (see arithmetic function).
These prime factor counting functions have many important number theoretic relations.
An asymptotic series for the average order of
is related to divisor sums over the Möbius function and the divisor function including the next sums.
[3] The characteristic function of the primes can be expressed by a convolution with the Möbius function:[4] A partition-related exact identity for
is expanded by in terms of the infinite q-Pochhammer symbol and the restricted partition functions
into an odd (even) number of distinct parts.
This is closely related to the following partition identity.
is prime a lower bound on the value of the function is
are respectively computed in Hardy and Wright as [10] [11] where
is defined by The sum of number of unitary divisors:
[12] (sequence A064608 in the OEIS) Other sums relating the two variants of the prime omega functions include [13] 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.
[14] To be completely precise, let the odd-indexed summatory function be defined as where
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
, 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
Now we can prove a short lemma in the following form which implies exact formulas for the expansions of the Dirichlet series over both
is a strongly additive arithmetic function defined such that its values at prime powers is given by
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 distribution of the distinct integer values of the differences
is regular in comparison with the semi-random properties of the component functions.
, define These cardinalities have a corresponding sequence of limiting densities
These densities are generated by the prime products With the absolute constant
satisfy Compare to the definition of the prime products defined in the last section of [16] in relation to the Erdős–Kac theorem.