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.
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]
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
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 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 , define
These cardinalities have a corresponding sequence of limiting densities such that for
^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.
^This identity is proved in the article by Schmidt cited on this page below.
^Hoelscher, Zachary; Palsson, Eyvindur (2020-12-05). "Counting Restricted Partitions of Integers Into Fractions: Symmetry and Modes of the Generating Function and a Connection to ω(t)". The PUMP Journal of Undergraduate Research. 3: 277–307. arXiv:2011.14502. ISSN 2576-3725.
^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.
^See 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 .
^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.
^This identity is found in Section 27.4 of the NIST Handbook of Mathematical Functions.
^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.