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). 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.

Properties and relationsEdit

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.  

 

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 planeEdit

A continuation of   has been found, though it is not analytic everywhere.[7] Note that the normalized   function is used.

 

Average order and summatory functionsEdit

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 again 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 functionEdit

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'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:

 

Example II: Summatory functions for so-termed factorial moments of  Edit

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 seriesEdit

A known Dirichlet series involving   and the Riemann zeta function is given by [13]

 

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 functionsEdit

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 alsoEdit

NotesEdit

  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.

ReferencesEdit

  • G. H. Hardy and E. M. Wright (2006). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press.
  • H. L. Montgomery and R. C. Vaughan (2007). Multiplicative number theory I. Classical theory (1st ed.). Cambridge University Press.
  • Schmidt, Maxie (2017). "Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions". arXiv:1712.00608 [math.NT].
  • Weisstein, Eric. "Distinct Prime Factors". MathWorld. Retrieved 22 April 2018.

External linksEdit