It occurs to me that a stronger version of my related post the other day would be:

$$\pi(ab) \geq \pi(a) \cdot \pi(b)\ ,$$

which I believe holds for all naturals $a,b$ except for the two pairs $(5,7)$ and $(7,7)$.

My first thought on a proof was to point out that

$$\pi(ab)\sim\frac{ab}{\log(ab)}>\frac{ab}{\log(a)\log(b)}\sim \pi(a)\pi(b)\ ,$$

given $a>e$ and $b>a^{1-\log{a}}$ to rule out the small log problems, though I'm not sure that's necessary.

My question is whether that's enough to establish that the conjecture is always true for sufficiently large $ab$, or if it merely implies that it will tend towards being true in the limit. If this isn't enough and someone has a different way to go at it, I'd be interested to hear it.

And if this approach is valid, is it safe to change the conjecture to the stricter $\pi(ab)>\pi(a)\pi(b)$ (for large enough $ab$)?


From Properties of Some Functions Connected To Prime Numbers - Mincu et al.

Theorem 3.4: For $x,y\geq \sqrt{53}$, $\pi(x)\pi(y)\leq \pi(xy)$

Remark 6.6 The relation $π(xy) ≥ π(x)π(y)$ holds for all positive integers $x, y$ with the following three exceptions: x = 5, y = 7; x = 7, y = 5 and x = y = 7.

So note the additional exception of $(7,7)$.

The proof is similar to your reasoning, with some additional edge cases.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.