Prime omega function

In number theory, the prime omega functions $ω (n)$ and $Ω (n)$ count the number of prime factors of a natural number $n$ . The number of distinct prime factors is assigned to $ω (n)$ (little omega - sequence A001221 in the OEIS), while $Ω (n)$ (big omega - sequence A001222 in the OEIS) counts the total number of prime factors with multiplicity (see arithmetic function). That is, if we have a prime factorization of $n$ of the form $n = p_{1}^{α_{1}} p_{2}^{α_{2}} \dots p_{k}^{α_{k}}$ for distinct primes $p_{i}$ ( $1 \leq i \leq k$ ), then the prime omega functions are given by $ω (n) = k$ and $Ω (n) = α_{1} + α_{2} + \dots + α_{k}$ . These prime-factor-counting functions have many important number theoretic relations.

Properties and relations

The function $ω (n)$ is additive and $Ω (n)$ is completely additive. Little omega has the formula

$ω (n) = \sum_{p ∣ n} 1,$

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

Big omega has the formulas

$Ω (n) = \sum_{p^{α} ∣ n} 1 = \sum_{p^{α} ∥ n} α .$

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, $Ω (12) = Ω (2^{2} 3^{1}) = 3$ .

The omegas are related by the inequalities $ω (n) \leq Ω (n)$ and $2 ω (n) \leq d (n) \leq 2 Ω (n)$ , where $d (n)$ is the divisor-counting function. If $Ω (n) = ω (n)$ , then $n$ is squarefree and related to the Möbius function by

μ (n) = (- 1)^{ω (n)} = (- 1)^{Ω (n)} .

If $ω (n) = 1$ then $n$ is a prime power, and if $Ω (n) = 1$ then $n$ is prime.

An asymptotic series for the average order of $ω (n)$ is

\frac{1}{n} \sum_{k = 1}^{n} ω (k) \sim \log \log n + B_{1} + \sum_{k \geq 1} (\sum_{j = 0}^{k - 1} \frac{γ_{j}}{j!} - 1) \frac{(k - 1)!}{(\log n)^{k}},

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

The function $ω (n)$ is related to divisor sums over the Möbius function and the divisor function, including:

\sum_{d ∣ n} | μ (d) | = 2^{ω (n)}

is the number of unitary divisors. (sequence A034444 in the OEIS)

\sum_{d ∣ n} | μ (d) | k^{ω (d)} = (k + 1)^{ω (n)}

\sum_{r ∣ n} 2^{ω (r)} = d (n^{2})

\sum_{r ∣ n} 2^{ω (r)} d (\frac{n}{r}) = d^{2} (n)

\sum_{d ∣ n} (- 1)^{ω (d)} = \prod_{p^{α} | | n} (1 - α)

\sum_{\overset{1 \leq k \leq m}{(k, m) = 1}} gcd (k^{2} - 1, m_{1}) gcd (k^{2} - 1, m_{2}) = φ (n) \sum_{\overset{d_{1} ∣ m_{1}}{d_{2} ∣ m_{2}}} φ (gcd (d_{1}, d_{2})) 2^{ω (lcm (d_{1}, d_{2}))}, m_{1}, m_{2} odd, m = lcm (m_{1}, m_{2})

\sum_{\overset{1 \leq k \leq n}{\gcd (k, m) = 1}} 1 = n \frac{φ (m)}{m} + O (2^{ω (m)})

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

χ_{ℙ} (n) = (μ * ω) (n) = \sum_{d | n} ω (d) μ (n / d) .

A partition-related exact identity for $ω (n)$ is given by

ω (n) = \log_{2} [\sum_{k = 1}^{n} \sum_{j = 1}^{k} (\sum_{d ∣ k} \sum_{i = 1}^{d} p (d - j i)) s_{n, k} \cdot | μ (j) |],

where $p (n)$ is the partition function, $μ (n)$ 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.

Continuation to the complex plane

A continuation of $ω (n)$ has been found, though it is not analytic everywhere. Note that the normalized $sinc$ function $sinc (x) = \frac{\sin (π x)}{π x}$ is used.

ω (z) = \log_{2} (\sum_{n = 1}^{⌈ R e (z) ⌉} sinc (\prod_{m = 1}^{⌈ R e (z) ⌉ + 1} (n^{2} + n - m z)))

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

a = \frac{2}{c} + \frac{4}{c} + \dots + \frac{2 (b - 1)}{c} + \frac{2 b}{c}

where $a$ , $b$ , and $c$ are positive integers, and $a > b > c$ . The number of partitions is then given by $2^{ω (a)} - 2$ .

Average order and summatory functions

An average order of both $ω (n)$ and $Ω (n)$ is $\log \log n$ . When $n$ is prime a lower bound on the value of the function is $ω (n) = 1$ . Similarly, if $n$ is primorial then the function is as large as

$ω (n) \sim \frac{\log n}{\log \log n}$

on average order. When $n$ is a power of 2, then $Ω (n) = \log_{2} (n) .$

Asymptotics for the summatory functions over $ω (n)$ , $Ω (n)$ , and powers of $ω (n)$ are respectively

\begin{matrix} \sum_{n \leq x} ω (n) & = x \log \log x + B_{1} x + o (x) \\ \sum_{n \leq x} Ω (n) & = x \log \log x + B_{2} x + o (x) \\ \sum_{n \leq x} ω (n)^{2} & = x (\log \log x)^{2} + O (x \log \log x) \\ \sum_{n \leq x} ω (n)^{k} & = x (\log \log x)^{k} + O (x (\log \log x)^{k - 1}), k \in ℤ^{+}, \end{matrix}

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

B_{2} = B_{1} + \sum_{p prime} \frac{1}{p (p - 1)} \approx 1.0345061758.

The sum of number of unitary divisors is

$\sum_{n \leq x} 2^{ω (n)} = (x \log x) / ζ (2) + O (x)$ (sequence A064608 in the OEIS)

Other sums relating the two variants of the prime omega functions include

\sum_{n \leq x} {Ω (n) - ω (n)} = O (x),

and

# {n \leq x : Ω (n) - ω (n) > \sqrt{\log \log x}} = O (\frac{x}{(\log \log x)^{1 / 2}}) .

Example I: A modified summatory function

In this example we suggest a variant of the summatory functions $S_{ω} (x) := \sum_{n \leq x} ω (n)$ 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 $S_{ω} (x)$ provided in the formulas in the main subsection of this article above.

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

S_{odd} (x) := \sum_{n \leq x} ω (n) [n odd],

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

S_{odd} (x) = \frac{x}{2} \log \log x + \frac{(2 B_{1} - 1) x}{4} + {\frac{x}{4}} - [x \equiv 2, 3 mod 4] + O (\frac{x}{\log x}) .

The proof of this result follows by first observing that

ω (2 n) = {\begin{matrix} ω (n) + 1, & if n is odd; \\ ω (n), & if n is even, \end{matrix}

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

\begin{matrix} S_{ω} (x) & = S_{odd} (x) + \sum_{n \leq ⌊ \frac{x}{2} ⌋} ω (2 n) \\ = S_{odd} (x) + \sum_{n \leq ⌊ \frac{x}{4} ⌋} (ω (4 n) + ω (4 n + 2)) \\ = S_{odd} (x) + \sum_{n \leq ⌊ \frac{x}{4} ⌋} (ω (2 n) + ω (2 n + 1) + 1) \\ = S_{odd} (x) + S_{ω} (⌊ \frac{x}{2} ⌋) + ⌊ \frac{x}{4} ⌋ . \end{matrix}

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

ω (n) {ω (n) - 1},

by estimating the product of these two component omega functions as

ω (n) {ω (n) - 1} = \sum_{\overset{p q ∣ n}{\overset{p \neq q}{p, q prime}}} 1 = \sum_{\overset{p q ∣ n}{p, q prime}} 1 - \sum_{\overset{p^{2} ∣ n}{p prime}} 1.

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

Dirichlet series

A known Dirichlet series involving $ω (n)$ and the Riemann zeta function is given by

\sum_{n \geq 1} \frac{2^{ω (n)}}{n^{s}} = \frac{ζ^{2} (s)}{ζ (2 s)}, ℜ (s) > 1.

We can also see that

\sum_{n \geq 1} \frac{z^{ω (n)}}{n^{s}} = \prod_{p} (1 + \frac{z}{p^{s} - 1}), | z | < 2, ℜ (s) > 1,

\sum_{n \geq 1} \frac{z^{Ω (n)}}{n^{s}} = \prod_{p} {(1 - \frac{z}{p^{s}})}^{- 1}, | z | < 2, ℜ (s) > 1,

The function $Ω (n)$ is completely additive, where $ω (n)$ 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 $ω (n)$ and $Ω (n)$ :

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

\sum_{n \geq 1} \frac{f (n)}{n^{s}} = ζ (s) \times \sum_{p p r i m e} (1 - p^{- s}) \cdot \sum_{n \geq 1} f_{0} (p, n) p^{- n s}, ℜ (s) > min (1, σ_{f}) .

Proof. We can see that

\sum_{n \geq 1} \frac{u^{f (n)}}{n^{s}} = \prod_{p p r i m e} (1 + \sum_{n \geq 1} u^{f_{0} (p, n)} p^{- n s}) .

This implies that

\begin{matrix} \sum_{n \geq 1} \frac{f (n)}{n^{s}} & = \frac{d}{d u} [\prod_{p p r i m e} (1 + \sum_{n \geq 1} u^{f_{0} (p, n)} p^{- n s})] |_{u = 1} = \prod_{p} (1 + \sum_{n \geq 1} p^{- n s}) \times \sum_{p} \frac{\sum_{n \geq 1} f_{0} (p, n) p^{- n s}}{1 + \sum_{n \geq 1} p^{- n s}} \\ = ζ (s) \times \sum_{p p r i m e} (1 - p^{- s}) \cdot \sum_{n \geq 1} f_{0} (p, n) p^{- n s}, \end{matrix}

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 $ℜ (s) > 1$ ,

\begin{matrix} D_{ω} (s) & := \sum_{n \geq 1} \frac{ω (n)}{n^{s}} = ζ (s) P (s) \\ = ζ (s) \times \sum_{n \geq 1} \frac{μ (n)}{n} \log ζ (n s) \\ D_{Ω} (s) & := \sum_{n \geq 1} \frac{Ω (n)}{n^{s}} = ζ (s) \times \sum_{n \geq 1} P (n s) \\ = ζ (s) \times \sum_{n \geq 1} \frac{ϕ (n)}{n} \log ζ (n s) \\ D_{h} (s) & := \sum_{n \geq 1} \frac{h (n)}{n^{s}} = ζ (s) \log ζ (s) \\ = ζ (s) \times \sum_{n \geq 1} \frac{ε (n)}{n} \log ζ (n s), \end{matrix}

where $P (s)$ is the prime zeta function, $h (n) = \sum_{p^{k} | n} \frac{1}{k} = \sum_{p^{k} | | n} H_{k}$ where $H_{k}$ is the $k$ -th harmonic number and $ε$ is the identity for the Dirichlet convolution, $ε (n) = ⌊ \frac{1}{n} ⌋$ .

The distribution of the difference of prime omega functions

The distribution of the distinct integer values of the differences $Ω (n) - ω (n)$ is regular in comparison with the semi-random properties of the component functions. For $k \geq 0$ , define

N_{k} (x) := # ({n \in ℤ^{+} : Ω (n) - ω (n) = k} \cap [1, x]) .

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

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

These densities are generated by the prime products

\sum_{k \geq 0} d_{k} \cdot z^{k} = \prod_{p} (1 - \frac{1}{p}) (1 + \frac{1}{p - z}) .

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

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

Compare to the definition of the prime products defined in the last section of in relation to the Erdős–Kac theorem.

Notes

↑ This inequality is given in Section 22.13 of Hardy and Wright.
↑ S. R. Finch, Two asymptotic series, Mathematical Constants II, Cambridge Univ. Press, pp. 21-32,
↑ 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 is suggested as an exercise in Apostol's book. Namely, we write $f = μ * ω$ where $f (n) = \sum_{d | n} μ (n / d) \sum_{r | d} (π (r) - π (r - 1))$ . We can form the Dirichlet series over $f$ as $D_{f} (s) := \sum_{n \geq 1} \frac{f (n)}{n^{s}} = P (s),$ where $P (s)$ is the prime zeta function. Then it becomes obvious to see that $f (n) = π (n) - π (n - 1) = χ_{ℙ} (n)$ is the indicator function of the primes.
↑ 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)
↑ 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. doi:10.46787/pump.v3i0.2428. ISSN 2576-3725.
↑ 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. doi:10.46787/pump.v3i0.2428. 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 $\sum_{n \leq x} ω (n)^{k}$ for any $k \geq 2$ by considering the summatory functions of the factorial moments of the form $\sum_{n \leq x} \frac{[ω (n)]!}{[ω (n) - m]!}$ for more general cases of $m \geq 2$ .
↑ Cohen, Eckford (1960). "The Number of Unitary Divisors of an Integer". The American Mathematical Monthly. 67 (9): 879–880. doi:10.2307/2309455. ISSN 0002-9890. JSTOR 2309455.
↑ 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.
↑ 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.

References

G. H. Hardy; E. M. Wright (2006). An Introduction to the Theory of Numbers (6th ed.). Oxford University Press.
H. L. Montgomery; 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 links

[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 $f = μ * ω$ where $f (n) = \sum_{d | n} μ (n / d) \sum_{r | d} (π (r) - π (r - 1))$ . We can form the Dirichlet series over $f$ as $D_{f} (s) := \sum_{n \geq 1} \frac{f (n)}{n^{s}} = P (s),$ where $P (s)$ is the prime zeta function. Then it becomes obvious to see that $f (n) = π (n) - π (n - 1) = χ_{ℙ} (n)$ 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] 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. doi:10.46787/pump.v3i0.2428. ISSN 2576-3725.

[8] 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. doi:10.46787/pump.v3i0.2428. ISSN 2576-3725.

[9] 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.

[10] See Sections 22.10 and 22.11 for reference and explicit derivations of these asymptotic estimates.

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

[12] Cohen, Eckford (1960). "The Number of Unitary Divisors of an Integer". The American Mathematical Monthly. 67 (9): 879–880. doi:10.2307/2309455. ISSN 0002-9890. JSTOR 2309455.

[13] Hardy and Wright Chapter 22.11.

[14] 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.

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

[16] 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.