Prime omega function

Prime omega function
In number theory, the prime omega functions and count the number of prime factors of a natural number Thereby (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). That is, 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.
Properties and relations
The function is additive and is completely additive.
If divides at least once we count it only once, e.g. .
If divides times then we count the exponents, e.g. . As usual, means is the exact power of dividing .
If then is squarefree and related to the Möbius function by
If then is a prime number.
It is known that the average order of the divisor function satisfies .
Like many arithmetic functions there is no explicit formula for or but there are approximations.
An asymptotic series for the average order of is given by
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 the next sums.
The characteristic function of the primes can be expressed by a convolution with the
Möbius function:
A partition-related exact identity for is given by
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.
Continuation to the complex plane
A continuation of has been found, though it is not analytic everywhere. Note that the normalized function is used.
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
.
Asymptotics for the summatory functions over and
are respectively computed in Hardy and Wright as
where is the Mertens constant and the constant is defined by
Other sums relating the two variants of the prime omega functions include
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.
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
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 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 , 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.
See also
Additive function
Arithmetic function
Erdős–Kac theorem
Omega function (disambiguation)
Prime number
Square-free integer
Notes
References
External links
OEIS Wiki for related sequence numbers and tables
OEIS Wiki on Prime Factors
Number theory
Prime numbers
Additive functions
Integer sequences