?

Number Theory

Free Version

Upgrade subject to access all content

Difficult

Distribution of Prime-Divisor-Counting Function, True/False

NUMTH-SB1OFS

Denote by $\omega(n)$ the number of prime divisors of $n$, and denote by $\Omega(n)$ the number of prime-power divisors of $n$.

If $n=p_1^{e_1}\cdots p_r^{e_r}$ is the factorization of $n$, then $\omega(n)=r$ and $\Omega(n)=e_1+\cdots + e_r$.

Denote by $\Phi(x)$ the cumulative distribution function of the standard normal distribution $\mathcal{N}(0, 1^2)$.

Which of the following statements are true? Select ALL that apply.

A

$\sum\limits_{n\leq x} \omega(n) =O(x\log\log x)$.

B

$\sum\limits_{n\leq x} \Omega(n) \neq O(x\log\log x)$.

C

$\lim\limits_{x\rightarrow\infty} \frac1x \# \{ n\leq x : \omega(n)-\log\log x \leq y\sqrt{\log\log x} \} = \Phi(y)$.

D

$\lim\limits_{x\rightarrow\infty} \frac1x \# \{ n\leq x : \Omega(n)-\log\log x \leq y \sqrt{\log\log x} \}=\Phi(y)$.