Prime omega function

In number theory, the prime omega functions ${\displaystyle \omega (n)}$ and ${\displaystyle \Omega (n)}$ count the number of prime factors of a natural number ${\displaystyle n.}$ Thereby ${\displaystyle \omega (n)}$ (little omega) counts each distinct prime factor, whereas the related function ${\displaystyle \Omega (n)}$ (big omega) counts the total number of prime factors of ${\displaystyle n,}$ honoring their multiplicity (see arithmetic function). For example, if we have a prime factorization of ${\displaystyle n}$ of the form ${\displaystyle n=p_{1}^{\alpha _{1}}p_{2}^{\alpha _{2}}\cdots p_{k}^{\alpha _{k}}}$ for distinct primes ${\displaystyle p_{i}}$ (${\displaystyle 1\leq i\leq k}$), then the respective prime omega functions are given by ${\displaystyle \omega (n)=k}$ and ${\displaystyle \Omega (n)=\alpha _{1}+\alpha _{2}+\cdots +\alpha _{k}}$. These prime factor counting functions have many important number theoretic relations.

Properties and relations

The function ${\displaystyle \omega (n)}$ is additive and ${\displaystyle \Omega (n)}$ is completely additive.

${\displaystyle \omega (n)=\sum _{p\mid n}1}$

If ${\displaystyle p}$ divides ${\displaystyle n}$ at least once we count it only once, e.g. ${\displaystyle \omega (12)=\omega (2^{2}3)=2}$

${\displaystyle \Omega (n)=\sum _{p^{\alpha }\mid n}\alpha }$

If ${\displaystyle p}$ divides ${\displaystyle n}$ ${\displaystyle \alpha }$ times then we count the exponents, e.g. ${\displaystyle \Omega (12)=\Omega (2^{2}3^{1})=3}$

${\displaystyle \Omega (n)\geq \omega (n)}$

If ${\displaystyle \Omega (n)=\omega (n)}$ then ${\displaystyle n}$ is squarefree and related to the Möbius function by

${\displaystyle \mu (n)=(-1)^{\omega (n)}=(-1)^{\Omega (n)}}$

If ${\displaystyle \Omega (n)=1}$ then ${\displaystyle n}$ is a prime number.

It is known that the average order of the divisor function satisfies ${\displaystyle 2^{\omega (n)}\leq d(n)\leq 2^{\Omega (n)}}$.[1]

Like many arithmetic functions there is no explicit formula for ${\displaystyle \Omega (n)}$ or ${\displaystyle \omega (n)}$ but there are approximations.

An asymptotic series for the average order of ${\displaystyle \omega (n)}$ is given by [2]

${\displaystyle {\frac {1}{n}}\sum \limits _{k=1}^{n}\omega (k)\sim \log \log n+B_{1}+\sum _{k\geq 1}\left(\sum _{j=0}^{k-1}{\frac {\gamma _{j}}{j!}}-1\right){\frac {(k-1)!}{(\log n)^{k}}},}$

where ${\displaystyle B_{1}\approx 0.26149721}$ is the Mertens constant and ${\displaystyle \gamma _{j}}$ are the Stieltjes constants.

The function ${\displaystyle \omega (n)}$ is related to divisor sums over the Möbius function and the divisor function including the next sums.[3]

${\displaystyle \sum _{d\mid n}|\mu (d)|=2^{\omega (n)}}$

${\displaystyle \sum _{r\mid n}2^{\omega (r)}=d(n^{2})}$

${\displaystyle \sum _{r\mid n}2^{\omega (r)}d\left({\frac {n}{r}}\right)=d^{2}(n)}$

${\displaystyle \sum _{\stackrel {1\leq k\leq m}{(k,m)=1}}\gcd(k^{2}-1,m_{1})\gcd(k^{2}-1,m_{2})=\varphi (n)\sum _{\stackrel {d_{1}\mid m_{1}}{d_{2}\mid m_{2}}}\varphi (\gcd(d_{1},d_{2}))2^{\omega (\operatorname {lcm} (d_{1},d_{2}))},\ m_{1},m_{2}{\text{ odd}},m=\operatorname {lcm} (m_{1},m_{2})}$

${\displaystyle \sum _{\stackrel {1\leq k\leq n}{\operatorname {gcd} (k,m)=1}}\!\!\!\!1=n{\frac {\varphi (m)}{m}}+O\left(2^{\omega (m)}\right)}$

The characteristic function of the primes can be expressed by a convolution with the Möbius function [4]:

${\displaystyle \chi _{\mathbb {P} }(n)=(\mu \ast \omega )(n)=\sum _{d|n}\omega (d)\mu (n/d).}$

A partition-related exact identity for ${\displaystyle \omega (n)}$ is given by [5]

${\displaystyle \omega (n)=\log _{2}\left[\sum _{k=1}^{n}\sum _{j=1}^{k}\left(\sum _{d\mid k}\sum _{i=1}^{d}p(d-ji)\right)s_{n,k}\cdot |\mu (j)|\right],}$

where ${\displaystyle p(n)}$ is the partition function, ${\displaystyle \mu (n)}$ is the Möbius function, and the triangular sequence ${\displaystyle s_{n,k}}$ is expanded by

${\displaystyle s_{n,k}=[q^{n}](q;q)_{\infty }{\frac {q^{k}}{1-q^{k}}}=s_{o}(n,k)-s_{e}(n,k),}$

in terms of the infinite q-Pochhammer symbol and the restricted partition functions ${\displaystyle s_{o/e}(n,k)}$ which respectively denote the number of ${\displaystyle k}$'s in all partitions of ${\displaystyle n}$ into an odd (even) number of distinct parts.[6]

Average order and summatory functions

An average order of both ${\displaystyle \omega (n)}$ and ${\displaystyle \Omega (n)}$ is ${\displaystyle \log \log n}$. When ${\displaystyle n}$ is prime a lower bound on the value of the function is ${\displaystyle \omega (n)=1}$. Similarly, if ${\displaystyle n}$ is primorial then the function is as large as ${\displaystyle \omega (n)\sim {\frac {\log n}{\log \log n}}}$ on average order. When ${\displaystyle n}$ is a power of 2, then ${\displaystyle \Omega (n)\sim {\frac {\log n}{\log 2}}}$ .[7]

Asymptotics for the summatory functions over ${\displaystyle \omega (n)}$, ${\displaystyle \Omega (n)}$, and ${\displaystyle \omega (n)^{2}}$ are respectively computed in Hardy and Wright as [8] [9]

${\displaystyle {\begin{aligned}\sum _{n\leq x}\omega (n)&=x\log \log x+B_{1}x+o(x)\\\sum _{n\leq x}\Omega (n)&=x\log \log x+B_{2}x+o(x)\\\sum _{n\leq x}\omega (n)^{2}&=x(\log \log x)^{2}+O(x\log \log x)\\\sum _{n\leq x}\omega (n)^{k}&=x(\log \log x)^{k}+O(x(\log \log x)^{k-1}),k\in \mathbb {Z} ^{+},\end{aligned}}}$

where ${\displaystyle B_{1}}$ is again the Mertens constant and the constant ${\displaystyle B_{2}}$ is defined by

${\displaystyle B_{2}=B_{1}+\sum _{p{\text{ prime}}}{\frac {1}{p(p-1)}}.}$

Other sums relating the two variants of the prime omega functions include [10]

${\displaystyle \sum _{n\leq x}\left\{\Omega (n)-\omega (n)\right\}=O(x),}$

and

${\displaystyle \#\left\{n\leq x:\Omega (n)-\omega (n)>{\sqrt {\log \log x}}\right\}=O\left({\frac {x}{(\log \log x)^{1/2}}}\right).}$

Example I: A modified summatory function

In this example we suggest a variant of the summatory functions ${\displaystyle S_{\omega }(x):=\sum _{n\leq x}\omega (n)}$ estimated in the above results for sufficiently large ${\displaystyle x}$. We then prove an asymptotic formula for the growth of this modified summatory function derived from the asymptotic estimate of ${\displaystyle S_{\omega }(x)}$ provided in the formulas in the main subsection of this article above.[11]

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

${\displaystyle S_{\operatorname {odd} }(x):=\sum _{n\leq x}\omega (n)[n{\text{ odd}}]_{\delta },}$

where ${\displaystyle [\cdot ]_{\delta }}$ denotes Iverson's convention. Then we have that

${\displaystyle S_{\operatorname {odd} }(x)={\frac {x}{2}}\log \log x+{\frac {(2B_{1}-1)x}{4}}+\left\{{\frac {x}{4}}\right\}-\left[x\equiv 2,3{\bmod {4}}\right]_{\delta }+O\left({\frac {x}{\log x}}\right).}$

The proof of this result follows by first observing that

${\displaystyle \omega (2n)={\begin{cases}\omega (n)+1,&{\text{if }}n{\text{ is odd; }}\\\omega (n),&{\text{if }}n{\text{ is even,}}\end{cases}}}$

and then applying the asymptotic result from Hardy and Wright for the summatory function over ${\displaystyle \omega (n)}$, denoted by ${\displaystyle S_{\omega }(x):=\sum _{n\leq x}\omega (n)}$, in the following form:

${\displaystyle {\begin{aligned}S_{\omega }(x)&=S_{\operatorname {odd} }(x)+\sum _{n\leq \left\lfloor {\frac {x}{2}}\right\rfloor }\omega (2n)\\&=S_{\operatorname {odd} }(x)+\sum _{n\leq \left\lfloor {\frac {x}{4}}\right\rfloor }\left(\omega (4n)+\omega (4n+2)\right)\\&=S_{\operatorname {odd} }(x)+\sum _{n\leq \left\lfloor {\frac {x}{4}}\right\rfloor }\left(\omega (2n)+\omega (2n+1)+1\right)\\&=S_{\operatorname {odd} }(x)+S_{\omega }\left(\left\lfloor {\frac {x}{2}}\right\rfloor \right)+\left\lfloor {\frac {x}{4}}\right\rfloor .\end{aligned}}}$

Example II: Summatory functions for so-termed factorial moments of ${\displaystyle \omega (n)}$

The computations expanded in Chapter 22.11 of Hardy and Wright provide asymptotic estimates for the summatory function

${\displaystyle \omega (n)\left\{\omega (n)-1\right\},}$

by estimating the product of these two component omega functions as

${\displaystyle \omega (n)\left\{\omega (n)-1\right\}=\sum _{\stackrel {pq\mid n}{\stackrel {p\neq q}{p,q{\text{ prime}}}}}1=\sum _{\stackrel {pq\mid n}{p,q{\text{ prime}}}}1-\sum _{\stackrel {p^{2}\mid n}{p{\text{ prime}}}}1.}$

We can similarly calculate asymptotic formulas more generally for the related summatory functions over so-termed factorial moments of the function ${\displaystyle \omega (n)}$.

Dirichlet series

A known Dirichlet series involving ${\displaystyle \omega (n)}$ and the Riemann zeta function is given by [12]

${\displaystyle \sum _{n\geq 1}{\frac {2^{\omega (n)}}{n^{s}}}={\frac {\zeta ^{2}(s)}{\zeta (2s)}},\ \Re (s)>1.}$

The function ${\displaystyle \Omega (n)}$ is completely additive, where ${\displaystyle \omega (n)}$ 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 ${\displaystyle \omega (n)}$ and ${\displaystyle \Omega (n)}$:

Lemma. Suppose that ${\displaystyle f}$ is a strongly additive arithmetic function defined such that its values at prime powers is given by ${\displaystyle f(p^{\alpha }):=f_{0}(p,\alpha )}$, i.e., ${\displaystyle f(p_{1}^{\alpha _{1}}\cdots p_{k}^{\alpha _{k}})=f_{0}(p_{1},\alpha _{1})+\cdots +f_{0}(p_{k},\alpha _{k})}$ for distinct primes ${\displaystyle p_{i}}$ and exponents ${\displaystyle \alpha _{i}\geq 1}$. The Dirichlet series of ${\displaystyle f}$ is expanded by

${\displaystyle \sum _{n\geq 1}{\frac {f(n)}{n^{s}}}=\zeta (s)\times \sum _{p\mathrm {\ prime} }(1-p^{-s})\cdot \sum _{n\geq 1}f_{0}(p,n)p^{-ns},\Re (s)>\min(1,\sigma _{f}).}$

Proof. We can see that

${\displaystyle \sum _{n\geq 1}{\frac {u^{f(n)}}{n^{s}}}=\prod _{p\mathrm {\ prime} }\left(1+\sum _{n\geq 1}u^{f_{0}(p,n)}p^{-ns}\right).}$

This implies that

${\displaystyle {\begin{aligned}\sum _{n\geq 1}{\frac {f(n)}{n^{s}}}&={\frac {d}{du}}\left[\prod _{p\mathrm {\ prime} }\left(1+\sum _{n\geq 1}u^{f_{0}(p,n)}p^{-ns}\right)\right]{\Biggr |}_{u=1}=\prod _{p}\left(1+\sum _{n\geq 1}p^{-ns}\right)\times \sum _{p}{\frac {\sum _{n\geq 1}f_{0}(p,n)p^{-ns}}{1+\sum _{n\geq 1}p^{-ns}}}\\&=\zeta (s)\times \sum _{p\mathrm {\ prime} }(1-p^{-s})\cdot \sum _{n\geq 1}f_{0}(p,n)p^{-ns},\end{aligned}}}$

wherever the corresponding series and products are convergent. In the last equation, we have used the Euler product representation of the Riemann zeta function. ${\displaystyle \boxdot }$

The lemma implies that for ${\displaystyle \Re (s)>1}$,

${\displaystyle {\begin{aligned}D_{\omega }(s)&:=\sum _{n\geq 1}{\frac {\omega (n)}{n^{s}}}=\zeta (s)P(s)\\&\ =\zeta (s)\times \sum _{n\geq 1}{\frac {\mu (n)}{n}}\log \zeta (ns)\\D_{\Omega }(s)&:=\sum _{n\geq 1}{\frac {\Omega (n)}{n^{s}}}=\zeta (s)\times \sum _{n\geq 1}P(ns)\\&\ =\zeta (s)\times \sum _{n\geq 1}{\frac {\phi (n)}{n}}\log \zeta (ns)\\D_{\Omega \lambda }(s)&:=\sum _{n\geq 1}{\frac {\lambda (n)\Omega (n)}{n^{s}}}=\zeta (s)\log \zeta (s),\end{aligned}}}$

where ${\displaystyle P(s)}$ is the prime zeta function and ${\displaystyle \lambda (n)=(-1)^{\Omega (n)}}$ is the Liouville lambda function.

The distribution of the difference of prime omega functions

The distribution of the distinct integer values of the differences ${\displaystyle \Omega (n)-\omega (n)}$ is regular in comparison with the semi-random properties of the component functions. For ${\displaystyle k\geq 0}$, let the sets

${\displaystyle N_{k}(x):=\#\{n\leq x:\Omega (n)-\omega (n)=k\}.}$

These sets have a corresponding sequence of limiting densities ${\displaystyle d_{k}}$ such that for ${\displaystyle x\geq 2}$

${\displaystyle N_{k}(x)=d_{k}\cdot x+O\left(\left({\frac {3}{4}}\right)^{k}{\sqrt {x}}(\log x)^{\frac {4}{3}}\right).}$

These densities are generated by the prime products

${\displaystyle \sum _{k\geq 0}d_{k}\cdot z^{k}=\prod _{p}\left(1-{\frac {1}{p}}\right)\left(1+{\frac {1}{p-z}}\right).}$

With the absolute constant ${\displaystyle {\hat {c}}:={\frac {1}{4}}\times \prod _{p>2}\left(1-{\frac {1}{(p-1)^{2}}}\right)^{-1}}$, the densities ${\displaystyle d_{k}}$ satisfy

${\displaystyle d_{k}={\hat {c}}\cdot 2^{-k}+O(5^{-k}).}$

Compare to the definition of the prime products defined in the last section of [13] 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

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,
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 ${\displaystyle f=\mu \ast \omega }$ where ${\displaystyle f(n)=\sum _{d|n}\mu (n/d)\sum _{r|d}\left(\pi (r)-\pi (r-1)\right)}$. We can form the Dirichlet series over ${\displaystyle f}$ as ${\displaystyle D_{f}(s):=\sum _{n\geq 1}{\frac {f(n)}{n^{s}}}=P(s),}$ where ${\displaystyle P(s)}$ is the prime zeta function. Then it becomes obvious to see that ${\displaystyle f(n)=\pi (n)-\pi (n-1)=\chi _{\mathbb {P} }(n)}$ 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. 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.
8. See Sections 22.10 and 22.11 for reference and explicit derivations of these asymptotic estimates.
9. 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 ${\displaystyle \sum _{n\leq x}\omega (n)^{k}}$ for any ${\displaystyle k\geq 2}$ by considering the summatory functions of the factorial moments of the form ${\displaystyle \sum _{n\leq x}{\frac {\left[\omega (n)\right]!}{\left[\omega (n)-m\right]!}}}$ for more general cases of ${\displaystyle m\geq 2}$.
10. Hardy and Wright Chapter 22.11.
11. 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.
12. This identity is found in Section 27.4 of the NIST Handbook of Mathematical Functions.
13. Rényi, A.; Turán, P. (1958). "On a theorem of Erdös-Kac" (PDF). Acta Arithmetica. 4 (1): 71–84.

References

• 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. "Factorization Theorems for Hadamard Products and Higher-Order Derivatives of Lambert Series Generating Functions". arXiv:1712.00608.
• Weisstein, Eric. "Distinct Prime Factors". MathWorld. Retrieved 22 April 2018.

External links

• OEIS Wiki for related sequence numbers and tables
• OEIS Wiki on Prime Factors