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 .[1]

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 [2]

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.[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 given by [5]

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.[6]

Continuation to the complex plane

A continuation of has been found, though it is not analytic everywhere.[7] 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 .[8]

Asymptotics for the summatory functions over , , and are respectively computed in Hardy and Wright as [9] [10]

where is the Mertens constant and the constant is defined by

Other sums relating the two variants of the prime omega functions include [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.[12]

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 [13]

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 , 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 [14] in relation to the Erdős–Kac theorem.

See also

Notes

  1. ^ This inequality is given in Section 22.13 of Hardy and Wright.
  2. ^ S. R. Finch, Two asymptotic series, Mathematical Constants II, Cambridge Univ. Press, pp. 21-32, [1]
  3. ^ 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.
  4. ^ 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.
  5. ^ This identity is proved in the article by Schmidt cited on this page below.
  6. ^ This triangular sequence also shows up prominently in the Lambert series factorization theorems proved by Merca and Schmidt (2017–2018)
  7. ^ Z. Hoelscher & E. Palsson, Counting restricted partitions of integers into fractions: symmetry and modes of the generating function and a connection to , The PUMP Journal of Undergraduate Research, 3 (2020), 277-307. [2]
  8. ^ 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.
  9. ^ See Sections 22.10 and 22.11 for reference and explicit derivations of these asymptotic estimates.
  10. ^ 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 .
  11. ^ Hardy and Wright Chapter 22.11.
  12. ^ 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.
  13. ^ This identity is found in Section 27.4 of the NIST Handbook of Mathematical Functions.
  14. ^ Rényi, A.; Turán, P. (1958). "On a theorem of Erdös-Kac" (PDF). Acta Arithmetica. 4 (1): 71–84. doi:10.4064/aa-4-1-71-84.

References