Prime omega function

In number theory, the prime omega functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} and $\Omega (n)$ count the number of prime factors of a natural number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n.} Thereby $\omega (n)$ (little omega) counts each distinct prime factor, whereas the related function $\Omega (n)$ (big omega) counts the total number of prime factors of $n,$ honoring their multiplicity (see arithmetic function). That is, 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 the 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.

Properties and relations

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

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

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

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

If $p$ divides $n$ $\alpha \geq 1$ times then we count the exponents, e.g. $\Omega (12)=\Omega (2^{2}3^{1})=3$. As usual, $p^{\alpha }\parallel n$ means $\alpha$ is the exact power of $p$ dividing $n$.

$\Omega (n)\geq \omega (n)$

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

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

If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \omega (n)=1} then $n$ is a prime power, and if $\Omega (n)=1$ then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is a prime number.

It is known that the divisor function satisfies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{\omega(n)} \leq d(n) \leq 2^{\Omega(n)}} .

Like many arithmetic functions there is no explicit formula for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega(n)} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n) } but there are approximations.

An asymptotic series for the average order of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_1 \approx 0.26149721} is the Mertens constant and $\gamma _{j}$ are the Stieltjes constants.

The function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} is related to divisor sums over the Möbius function and the divisor function including the next sums.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{d\mid n} |\mu(d)| = 2^{\omega(n)} } is the number of unitary divisors. OEIS: A034444

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{d\mid n} |\mu(d)| k^{\omega(d)} = (k+1)^{\omega(n)} }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{r\mid n} 2^{\omega(r)} = d(n^2) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{r\mid n} 2^{\omega(r)} d\left(\frac{n}{r}\right) = d^2(n) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{d\mid n} (-1)^{\omega(d)} = \prod\limits_{p^{\alpha}||n} (1-\alpha)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{\stackrel{1\le k\le 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) }

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

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi_{\mathbb{P}}(n) = (\mu \ast \omega)(n) = \sum_{d|n} \omega(d) \mu(n/d). }

A partition-related exact identity for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(n)} is the partition function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(n)} is the Möbius function, and the triangular sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_{n,k}} is expanded by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s_{o/e}(n, k)} which respectively denote the number of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} 's in all partitions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} into an odd (even) number of distinct parts.

Continuation to the complex plane

A continuation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} has been found, though it is not analytic everywhere. Note that the normalized Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \operatorname {sinc} } function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x} } is used.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a= \frac{2}{c} + \frac{4}{c} + \ldots + \frac{2(b-1)}{c} + \frac{2b}{c} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a } , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b } , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c } are positive integers, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a > b > c } . The number of partitions is then given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2^{\omega(a)} - 2 } .

Average order and summatory functions

An average order of both Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega(n)} is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \log\log n} . When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is prime a lower bound on the value of the function is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n) = 1} . Similarly, if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is primorial then the function is as large as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n) \sim \frac{\log n}{\log\log n}} on average order. When Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is a power of 2, then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega(n) \sim \frac{\log n}{\log 2}} .

Asymptotics for the summatory functions over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega(n)} , and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)^2} are respectively computed in Hardy and Wright as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_1 \approx 0.2614972128} is the Mertens constant and the constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_2} is defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle B_2 = B_1 + \sum_{p\text{ prime}} \frac{1}{p(p-1)} \approx 1.0345061758.}

The sum of number of unitary divisors:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n \le x} 2^{\omega(n)} =(x \log x)/\zeta(2) + O(x)} (sequence A064608 in the OEIS)

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n \leq x} \left\{\Omega(n) - \omega(n)\right\} = O(x), }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\omega}(x) := \sum_{n \leq x} \omega(n)} estimated in the above results for sufficiently large Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} . We then prove an asymptotic formula for the growth of this modified summatory function derived from the asymptotic estimate of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\omega}(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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\operatorname{odd}}(x) := \sum_{n \leq x} \omega(n) [n\text{ odd}], }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [\cdot]} denotes Iverson bracket. Then we have that

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} , denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_{\omega}(x) := \sum_{n \leq x} \omega(n)} , in the following form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align} }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n) \left\{\omega(n)-1\right\},}

by estimating the product of these two component omega functions as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} .

Dirichlet series

A known Dirichlet series involving Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} and the Riemann zeta function is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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,}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega(n)} is completely additive, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(n)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega(n)} :

Lemma. Suppose that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is a strongly additive arithmetic function defined such that its values at prime powers is given by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(p^{\alpha}) := f_0(p, \alpha)} , i.e., Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i} and exponents Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha_i \geq 1} . The Dirichlet series of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is expanded by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \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{align} }

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Re(s) > 1} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} 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{align} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(s)} is the prime zeta function, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h(n) = \sum_{p^k|n}{\frac{1}{k}} = \sum_{p^k||n}{H_{k}}} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_{k}} is the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} -th harmonic number and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} is the identity for the Dirichlet convolution, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega(n) - \omega(n)} is regular in comparison with the semi-random properties of the component functions. For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k \geq 0} , define

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_k} such that for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \geq 2}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{c} := \frac{1}{4} \times \prod_{p > 2} \left(1 - \frac{1}{(p-1)^2}\right)^{-1}} , the densities Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d_k} satisfy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 in relation to the Erdős–Kac theorem.

See also

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

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 (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

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