Prime omega function

The function \omega(n) is additive and \Omega(n) is completely additive. Little omega has the formula \omega(n)=\sum_{p\mid n} 1, where notation indicates that the sum is taken over all primes that divide , without multiplicity. For example, \omega(12)=\omega(2^2 3)=2. Big omega has the formulas \Omega(n) =\sum_{p^\alpha\mid n} 1 =\sum_{p^\alpha\parallel n}\alpha. The notation indicates that the sum is taken over all prime powers that divide , while indicates that the sum is taken over all prime powers that divide and such that is coprime to . For example, \Omega(12)=\Omega(2^2 3^1)=3. The omegas are related by the inequalities and , where is the divisor-counting function. If , then is squarefree and related to the Möbius function by :\mu(n) = (-1)^{\omega(n)} = (-1)^{\Omega(n)}. If \omega(n) = 1 then n is a prime power, and if \Omega(n)=1 then n is prime. An asymptotic series for the average order of \omega(n) is :\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 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: :\sum_{d\mid n} |\mu(d)| = 2^{\omega(n)} is the number of unitary divisors. :\sum_{d\mid n} |\mu(d)| k^{\omega(d)} = (k+1)^{\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_{d\mid n} (-1)^{\omega(d)} = \prod\limits_{p^{\alpha}||n} (1-\alpha) : \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\le k\le 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: : \chi_{\mathbb{P}}(n) = (\mu \ast \omega)(n) = \sum_{d|n} \omega(d) \mu(n/d). A partition-related exact identity for \omega(n) is given by :\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. ==Continuation to the complex plane==

A continuation of \omega(n) has been found, though it is not analytic everywhere. Note that the normalized \operatorname{sinc} function \operatorname{sinc}(x) = \frac{\sin(\pi x)}{\pi x} is used. :\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 :a= \frac{2}{c} + \frac{4}{c} + \ldots + \frac{2(b-1)}{c} + \frac{2b}{c} where a , b , and c are positive integers, and a > b > c . The number of partitions is then given by 2^{\omega(a)} - 2 . ==Average order and summatory functions==

An average order of both \omega(n) and \Omega(n) is \log\log n. When n is prime a lower bound on the value of the function is \omega(n) = 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) = \log_2(n). Asymptotics for the summatory functions over \omega(n), \Omega(n), and powers of \omega(n) are respectively :\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 B_1 \approx 0.2614972128 is the Mertens constant and the constant B_2 is defined by :B_2 = B_1 + \sum_{p\text{ prime}} \frac{1}{p(p-1)} \approx 1.0345061758. The sum of number of unitary divisors is \sum_{n \le x} 2^{\omega(n)} =(x \log x)/\zeta(2) + O(x) Other sums relating the two variants of the prime omega functions include :\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. 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}], where [\cdot] denotes Iverson bracket. 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] + 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{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 :\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). ==Dirichlet series==

A known Dirichlet series involving \omega(n) and the Riemann zeta function is given by :\sum_{n \geq 1} \frac{2^{\omega(n)}}{n^s} = \frac{\zeta^2(s)}{\zeta(2s)},\ \Re(s) > 1. We can also see that : \sum_{n \geq 1} \frac{z^{\omega(n)}}{n^s} = \prod_p \left(1 + \frac{z}{p^s-1}\right), |z| 1, : \sum_{n \geq 1} \frac{z^{\Omega(n)}}{n^s} = \prod_p \left(1 - \frac{z}{p^s}\right)^{-1}, |z| 1, The function \Omega(n) is completely additive, where \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 \omega(n) and \Omega(n): Lemma. Suppose that f is a strongly additive arithmetic function defined such that its values at prime powers is given by f(p^{\alpha}) := f_0(p, \alpha), i.e., 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 p_i and exponents \alpha_i \geq 1. The Dirichlet series of f is expanded by :\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 : \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 : \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 \Re(s) > 1, : \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 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 \varepsilon is the identity for the Dirichlet convolution, \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 \Omega(n) - \omega(n) is regular in comparison with the semi-random properties of the component functions. For k \geq 0, define :N_k(x) := \#(\{n \in \mathbb{Z}^{+}: \Omega(n) - \omega(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\left(\left(\frac{3}{4}\right)^k \sqrt{x} (\log x)^{\frac{4}{3}}\right). These densities are generated by the prime products :\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 \hat{c} := \frac{1}{4} \times \prod_{p > 2} \left(1 - \frac{1}{(p-1)^2}\right)^{-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. ==See also ==