Prime omega function

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

Known relations and identities

The functions $\omega (n)$ is additive and $\Omega (n)$ is completely additive. If $n$ is squarefree then $\omega (n)$ is related to the Möbius function by ^[1]

\mu (n)=(-1)^{\omega (n)}.

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

An asymptotic series for $\omega (n)$ is given by ^[3]

\omega (n)\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 $B_{1}\approx 0.26149721$ is the Mertens constant and $\gamma_j$ are the Stieltjes constants.

The function $\omega (n)$ is related to divisor sums over the Möbius function and the divisor function including the next sums .^[4]

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

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

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

\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})

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

A known Dirichlet series involving $\omega (n)$ and the Riemann zeta function is given by ^[5]

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

A partition-related exact identity for $\omega (n)$ is given by ^[6]

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

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 $s_{o/e}(n,k)$ which respectively denote the number of $k$ 's in all partitions of $n$ into an odd (even) number of distinct parts.^[7]

Average order and summatory functions

The average order of $\omega (n)$ is given by $\omega (n)\sim \log \log n$ and $\Omega (n)\sim \log \log n$ . When $p$ is prime a lower bound on the value of the function is $\omega (p)=1$ . Similarly, if $n$ is primorial then the function is as large as $\omega (n)\sim {\frac {\log n}{\log \log n}}$ on average order. When $n$ is a power of 2, then $\Omega (n)\sim {\frac {\log n}{\log 2}}$ .^[8]

Asymptotics for the summatory functions over $\omega (n)$ , $\Omega (n)$ , and $\omega (n)^{2}$ are respectively computed in Hardy and Wright as ^[9] ^[10]

{\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),\end{aligned}}

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

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 ^[11]

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

and

\#\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 $S_{\omega }(x):=\sum _{n\leq x}\omega (n)$ estimated in the above results for sufficiently large $x$ . We then prove an asymptotic formula for the growth of this modified summatory function derived from the asymptotic estimate of $S_{\omega }(x)$ 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

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

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

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

\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 $\omega (n)$ , denoted by $S_{\omega }(x):=\sum _{n\leq x}\omega (n)$ , in the following form:

{\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 $\omega (n)$

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

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

by estimating the product of these two component omega functions as

\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 $\omega (n)$ .

Notes

↑ This is obvious by the definition of the Möbius function.
↑ This inequality is given in Section 22.13 of Hardy and Wright.
↑ See equation (4) of the MathWorld reference. The truncation of this series to its main term is also cited below and proved in Section 22.10 of Hardy and Wright.
↑ 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.
↑ This identity is found in Section 27.4 of the NIST Handbook of Mathematical Functions.
↑ This identity is proved in the article by Schmidt cited on this page below.
↑ This triangular sequence also shows up prominently in the Lambert series factorization theorems proved by Merca and Schmidt (2017–2018)
↑ 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 $\sum _{n\leq x}\omega (n)^{k}$ for any $k\geq 2$ by considering the summatory functions of the factorial moments of the form $\sum _{n\leq x}{\frac {\left[\omega (n)\right]!}{\left[\omega (n)-m\right]!}}$ for more general cases of $m \geq 2$ .
↑ Hardy and Wright Chapter 22.11.
↑ 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.

References

G. H. Hardy and E. M. Wright (2006). An Introduction to the Theory of Numbers (6th ed.). Oxford 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

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[1] This is obvious by the definition of the Möbius function.

[2] This inequality is given in Section 22.13 of Hardy and Wright.

[3] See equation (4) of the MathWorld reference. The truncation of this series to its main term is also cited below and proved in Section 22.10 of Hardy and Wright.

[4] 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.

[5] This identity is found in Section 27.4 of the NIST Handbook of Mathematical Functions.

[6] This identity is proved in the article by Schmidt cited on this page below.

[7] This triangular sequence also shows up prominently in the Lambert series factorization theorems proved by Merca and Schmidt (2017–2018)

[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 $\sum _{n\leq x}\omega (n)^{k}$ for any $k\geq 2$ by considering the summatory functions of the factorial moments of the form $\sum _{n\leq x}{\frac {\left[\omega (n)\right]!}{\left[\omega (n)-m\right]!}}$ for more general cases of $m \geq 2$ .

[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.