Let $p_n$ be the $n$th prime and $a < p_n$ be a non-negative integer .
Let $f(a,p_n)$ be the number of integers $x$ such that:
$$a(p_{n-1}\#) < x < (a+1)(p_{n-1}\#) \text{ and } \gcd(x,p_n\#)=1$$
- where $p\#$ is the primorial for $p$
- and $\gcd(a,b)$ is the greatest common denominator between $a$ and $b$.
I am trying to show that if $\varphi(x)$ is the Euler Totient function, then:
$$f(a,p_n) > \varphi(p_{n-1}\#) - \varphi(p_{n-2}\#)$$
Assuming that my reasoning is correct (which it may not be), my goal is to state the following argument as clearly and concisely as possible.
Here is my argument:
(1) $f(a,p_n) < \varphi(p_{n-1}\#)$ since $\varphi(p_{n-1}\#) = $ the number of integers $x$ such that: $$a(p_{n-1}\#) < x < (a+1)(p_{n-1}\#) \text{ and } \gcd(x,p_{n-1}\#)=1$$
(2) Let $g(a,p_{n-1},p_n)$ be the number of integers $x$ such that: $$a(p_{n-1}\#) < x < (a+1)(p_{n-1}\#) \text{ and } \mathbb{lpf}(x)=p_n$$ where lpf(x) is the least prime factor for $x$.
(3) $g(a,p_{n-1},p_n) \le \varphi(p_{n-2}\#)$ since $g(a,p_{n-1},p_n) \le g(a,p_{n-1},p_{n-1}) = \varphi(p_{n-2}\#)$
(4) So the conclusion follows since $f(a,p_n) = \varphi(p_{n-1}\#) - g(a,p_{n-1},p_n)$
Is my argument valid? Is my argument as clear and concise as it could be? Is there an easier way to make the same point?
Edit: Adding details on (3) in response to a comment:
(3A): $g(a,p_{n-1},p_{n-1}) = \varphi(p_{n-2}\#)$
- $\mathbb{lpf(x)}=p_{n-1}$ if and only if $\exists y$ where $x=y p_{n-1}$ and:
$$a(p_{n-2}\#) < y < (a+1)(p_{n-2}\#) \text{ and } \mathbb{lpf}(y)\ge p_{n-1}$$
- The number of such integers $y$ is $\varphi(p_{n-2}\#)$. (I can provide more information on this if needed)
(3B): $g(a,p_{n-1},p_{n}) \le g(a,p_{n-1},p_{n-1})$
- $\mathbb{lpf(x)}=p_n$ if and only if $\exists y$ where $x=y p_n$ and:
$$a\left(\frac{p_{n-1}\#}{p_n}\right) < y < (a+1)\left(\frac{p_{n-1}\#}{p_n}\right) \text{ and } \mathbb{lpf}(y)\ge p_n$$
- The conclusion follows since:
$$a\left(\frac{p_{n-1}\#}{p_n}\right) < y < (a+1)\left(\frac{p_{n-1}\#}{p_n}\right) < a\left(\frac{p_{n-1}\#}{p_n}\right) + p_{n-2}\#$$
- which contains $\varphi(p_{n-2}\#)$ integers $x$ where $\mathbb{lpf}(x) \ge p_{n-1}$