Prime omega function

In number theory, the prime omega functions and count the number of prime factors of a natural number . The number of distinct prime factors is assigned to (little omega), while (big omega) counts the total number of prime factors with multiplicity (see arithmetic function). That is, if we have a prime factorization of of the form for distinct primes (), then the prime omega functions are given by and . These prime-factor-counting functions have many important number theoretic relations.

Properties and relations

The function is additive and is completely additive. Little omega has the formula

where notation indicates that the sum is taken over all primes that divide , without multiplicity. For example, .

Big omega has the formulas

The notation indicates that the sum is taken over all prime powers that divide , while indicates that the sum is taken over all prime powers that divide and such that is coprime to . For example, .

The omegas are related by the inequalities and , where is the divisor-counting function.This inequality is given in Section 22.13 of Hardy and Wright. If , then is squarefree and related to the Möbius function by

If then is a prime power, and if then is prime.

An asymptotic series for the average order of is S. R. Finch, Two asymptotic series, Mathematical Constants II, Cambridge Univ. Press, pp. 21-32,

where is the Mertens constant and are the Stieltjes constants.

The function is related to divisor sums over the Möbius function and the divisor function, including:Each of these started from the second identity in the list are cited individually on the pages Dirichlet convolutions of arithmetic functions, Menon's identity, and other formulas for Euler's totient function. The first identity is a combination of two known divisor sums cited in Section 27.6 of the NIST Handbook of Mathematical Functions.

is the number of unitary divisors.

The characteristic function of the primes can be expressed by a convolution with the Möbius function:This is suggested as an exercise in Apostol's book. Namely, we write where . We can form the Dirichlet series over as where is the prime zeta function. Then it becomes obvious to see that is the indicator function of the primes.

A partition-related exact identity for is given by This identity is proved in the article by Schmidt cited on this page below.

where is the partition function, is the Möbius function, and the triangular sequence is expanded by

in terms of the infinite q-Pochhammer symbol and the restricted partition functions which respectively denote the number of 's in all partitions of into an odd (even) number of distinct parts.This triangular sequence also shows up prominently in the Lambert series factorization theorems proved by Merca and Schmidt (2017–2018)

Continuation to the complex plane A continuation of has been found, though it is not analytic everywhere. Note that the normalized function is used.

This is closely related to the following partition identity. Consider partitions of the form

where , , and are positive integers, and . The number of partitions is then given by .

Average order and summatory functions

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 For references to each of these average order estimates see equations (3) and (18) of the MathWorld reference and Section 22.10-22.11 of Hardy and Wright.

Asymptotics for the summatory functions over , , and powers of are respectivelySee Sections 22.10 and 22.11 for reference and explicit derivations of these asymptotic estimates.Actually, the proof of the last result given in Hardy and Wright actually suggests a more general procedure for extracting asymptotic estimates of the moments for any by considering the summatory functions of the factorial moments of the form for more general cases of .

where is the Mertens constant and the constant is defined by

The sum of number of unitary divisors is

Other sums relating the two variants of the prime omega functions include Hardy and Wright Chapter 22.11.

and

Example I: A modified summatory function

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.N.b., this sum is suggested by work contained in an unpublished manuscript by the contributor to this page related to the growth of the Mertens function. Hence it is not just a vacuous and/or trivial estimate obtained for the purpose of exposition here.

To be completely precise, let the odd-indexed summatory function be defined as

where denotes Iverson bracket. 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:

Example II: Summatory functions for so-termed factorial moments of ω(n)

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 .

Dirichlet series

A known Dirichlet series involving and the Riemann zeta function is given by This identity is found in Section 27.4 of the NIST Handbook of Mathematical Functions.

We can also see that

The function is completely additive, where is strongly additive (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, where is the -th harmonic number and is the identity for the Dirichlet convolution, .

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 , define

These cardinalities 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.

Prime omega function

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