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 ${\displaystyle n.}$ The number of distinct prime factors is assigned to ${\displaystyle \omega (n)}$ (little omega), while ${\displaystyle \Omega (n)}$ (big omega) counts the total number of prime factors with multiplicity (see arithmetic function). That is, 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 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. Little omega has the formula

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

where notation p|n indicates that the sum is taken over all primes p that divide n, without multiplicity. For example, ${\displaystyle \omega (12)=\omega (2^{2}3)=2}$.

Big omega has the formulas

${\displaystyle \Omega (n)=\sum _{p^{\alpha }\mid n}1=\sum _{p^{\alpha }\parallel n}\alpha .}$

The notation p^α|n indicates that the sum is taken over all prime powers p^α that divide n, while p^α||n indicates that the sum is taken over all prime powers p^α that divide n and such that n / p^α is coprime to p^α. For example, ${\displaystyle \Omega (12)=\Omega (2^{2}3^{1})=3}$.

The omegas are related by the inequalities ω(n) ≤ Ω(n) and 2^ω(n) ≤ d(n) ≤ 2^Ω(n), where d(n) is the divisor-counting function.^[1] If Ω(n) = ω(n), then 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 power, and if ${\displaystyle \Omega (n)=1}$ then ${\displaystyle n}$ is prime.

An asymptotic series for the average order of ${\displaystyle \omega (n)}$ is ^[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:^[3]

${\displaystyle \sum _{d\mid n}|\mu (d)|=2^{\omega (n)}}$ is the number of unitary divisors. OEIS: A034444

${\displaystyle \sum _{d\mid n}|\mu (d)|k^{\omega (d)}=(k+1)^{\omega (n)}}$

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

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

${\displaystyle \sum _{d\mid n}(-1)^{\omega (d)}=\prod \limits _{p^{\alpha }||n}(1-\alpha )}$

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

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

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]

Continuation to the complex plane

A continuation of ${\displaystyle \omega (n)}$ has been found, though it is not analytic everywhere.^[7] Note that the normalized ${\displaystyle \operatorname {sinc} }$ function ${\displaystyle \operatorname {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}}$ is used.

${\displaystyle \omega (z)=\log _{2}\left(\sum _{n=1}^{\lceil Re(z)\rceil }\operatorname {sinc} \left(\prod _{m=1}^{\lceil Re(z)\rceil +1}\left(n^{2}+n-mz\right)\right)\right)}$

This is closely related to the following partition identity. Consider partitions of the form

${\displaystyle a={\frac {2}{c}}+{\frac {4}{c}}+\ldots +{\frac {2(b-1)}{c}}+{\frac {2b}{c}}}$

where ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$ are positive integers, and ${\displaystyle a>b>c}$. The number of partitions is then given by ${\displaystyle 2^{\omega (a)}-2}$. ^[8]

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 ${\displaystyle \Omega (n)=\log _{2}(n).}$^[9]

Asymptotics for the summatory functions over ${\displaystyle \omega (n)}$, ${\displaystyle \Omega (n)}$, and powers of ${\displaystyle \omega (n)}$ are respectively^[10][11]

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

where ${\displaystyle B_{1}\approx 0.2614972128}$ is the Mertens constant and the constant ${\displaystyle B_{2}}$ is defined by

${\displaystyle B_{2}=B_{1}+\sum _{p{\text{ prime}}}{\frac {1}{p(p-1)}}\approx 1.0345061758.}$

The sum of number of unitary divisors is

${\displaystyle \sum _{n\leq x}2^{\omega (n)}=(x\log x)/\zeta (2)+O(x)}$^[12] (sequence A064608 in the OEIS)

Other sums relating the two variants of the prime omega functions include ^[13]

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

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.^[14]

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

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

where ${\displaystyle [\cdot ]}$ denotes Iverson bracket. Then we have that

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

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 ω(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 ^[15]

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

We can also see that

${\displaystyle \sum _{n\geq 1}{\frac {z^{\omega (n)}}{n^{s}}}=\prod _{p}\left(1+{\frac {z}{p^{s}-1}}\right),|z|<2,\Re (s)>1,}$

${\displaystyle \sum _{n\geq 1}{\frac {z^{\Omega (n)}}{n^{s}}}=\prod _{p}\left(1-{\frac {z}{p^{s}}}\right)^{-1},|z|<2,\Re (s)>1,}$

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.

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

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

where ${\displaystyle P(s)}$ is the prime zeta function, ${\displaystyle h(n)=\sum _{p^{k}|n}{\frac {1}{k}}=\sum _{p^{k}||n}{H_{k}}}$ where ${\displaystyle H_{k}}$ is the ${\displaystyle k}$-th harmonic number and ${\displaystyle \varepsilon }$ is the identity for the Dirichlet convolution, ${\displaystyle \varepsilon (n)=\lfloor {\frac {1}{n}}\rfloor }$.

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 ${\displaystyle k\geq 0}$, define

${\displaystyle N_{k}(x):=\#(\{n\in \mathbb {Z} ^{+}:\Omega (n)-\omega (n)=k\}\cap [1,x]).}$

These cardinalities 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 ^[16] in relation to the Erdős–Kac theorem.

See also

• Additive function
• Arithmetic function
• Erdős–Kac theorem
• Omega function (disambiguation)
• Prime number
• Square-free integer