How can I prove this inequality regarding the prime numbers?

Question

The inequality in question is: $$\sum_{p|2n} (\lfloor {n \over p} \rfloor - 1)-\sum_{p_1p_2|2n\;p_1>p_2} \lfloor {n \over p_1p_2} \rfloor + \sum_{p_1p_2p_3|2n\;p_1>p_2>p_3} \lfloor {n \over p_1p_2p_3} \rfloor - \cdots + 1 > n - \pi(2n-2)$$ Where $\pi$ is the prime-counting function, and for $n$ arbitrarily large. Sorry to post this without context, but even if I showed you the context, it probably wouldn't be of much use to solving it.

So, if this is held to be true for all $n$ that are arbitrarily large, how to prove so?

EDIT: It seems that for when $n$ is a prime number, this inequality tends to be false, but, I'm only interested in the cases where $n$ is not prime. So, let me rephrase my question: How can we prove this inequality, for all composite values of $n$ that are arbitrarily large?

Answer 1

It should be easily enough falsifiable in the case where $n=pq$ with $p,q\to\infty$ and $p/q\to1$ as well: The left hand side is $\left(\frac1p+\frac1q+\frac12\right)n-3-\left(\frac1{2p}+\frac1{2q}+\frac1{pq}\right)n$ $=\left(\frac1{2p}+\frac1{2q}+\frac12\right)n+O(1)$ $\approx n\left(\frac12+\frac1{\sqrt{n}}\right)$ whereas the right hand side is $n-\frac{2n}{\ln(2n)}+o(n/\ln n)\approx n\left(1-\frac2{\ln(2n)}\right)$. So all that should be needed here is to choose $n$ large enough that the multiplier of $n$ on the LHS of your equation (i.e. $\left(\frac12+\frac1{\sqrt{n}}\right)$) is $\lt\frac23$, say, and so that the multiplier of $n$ on the RHS of your equation (i.e., $\left(1-\frac2{\ln(2n)}\right)$) is $\gt\frac45$.

For instance, suppose we take $n=197\cdot 199 = 39203$. Then the left hand side is $(197-1+199-1+19601-1)-(98+99+1)+1$ $=19797$, but the RHS is $39203-\pi(78404) = 39203-7695 = 31508$.

Answer 2

Let $$f(n)=\sum_{j\ge 0, p_1\ldots p_j |2n, p_i >p_{i+1}} (-1)^j \lfloor \frac{n}{p_1\ldots p_j}\rfloor$$ $$= \sum_{d | 2n}\mu(d) \lfloor n/d \rfloor=\sum_{m \le n} \sum_{d | \gcd(m,2n)} \mu(d)=\sum_{m\le n, \gcd(m,2n)=1} 1$$ If $n$ is odd and $\gcd(m,n)=1,m\le n$ then one of $m,n-m$ is odd and coprime with $2n$, thus $f(n)=\phi(n)/2$. If $n$ is even then $f(n)=\phi(n)$.

And your sum is

$$-\sum_{p | 2n} 1 +(n-f(n))+1= n+1-f(n)-\sum_{p | 2n} 1$$ Thus your claim is that $\pi(2n-2) \ge f(n)+\sum_{p | 2n} 1$ which is obviously false for $n$ prime large enough.