# What is this function, and what are it's properties?

I made a function that determines how "prime-y" a number is; if $f(x) = 1$ then $x \in primes$. The function is $$f(x) = \frac{\pi(x) - \#\{p \in primes : p<x \land p \space| \space x\}}{\pi(x)}$$ where the pound sign is use as the symbol of cardinality, and the vertical bar is used both as such that, and divides (context should be sufficient). There are several interesting properties of the function, so I'm wondering if anybody has explored this function at all in depth, and if so what is the real name of this function (calling the prime-y-ness or prime-like-ness of a number, as I have been doing, is very annoying and a little bit awkward)?

• You could also use $:$ to indicate such that. – Irregular User Jun 5 '16 at 23:55
• Please tell us a property of this function that you consider interesting. – KCd Jun 6 '16 at 0:34
• @KCd $lim_{x\rightarrow \infty}f(x) = 1$ – tox123 Sep 2 '17 at 16:51
• That is not really interesting since it is elementary to explain. If $x$ has prime factorization $p_1^{e_1}\cdots p_r^{e_r}$ ($p_i$ are distinct primes and $e_i$ are positive integers) then $f(x) = 1 - r/\pi(x)$ if $x$ is not prime and $f(x) = 1 - 1/\pi(x)$ if $x$ is prime. Since $p_i \geq 2$, we have $x \geq p_1\cdots p_r \geq 2^r$ so $r\leq \log_2 x$ and thus $0 < 1/\pi(x) \leq r/\pi(x) \leq \log_2(x)/\pi(x)$. It is elementary to show $\pi(x) \geq Cx/\ln x$ for a constant $C>0$, so $\log_2(x)/\pi(x) \leq (\log_2(x)\ln(x))/Cx$, which obviously tends to $0$ as $x\rightarrow \infty$. – KCd Sep 2 '17 at 18:01