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Let $ \pi(n) $ be the prime counting function, by "weak prime number theorem" I mean:

$$\lim_{n \to \infty}\frac{\sum_{k=1}^n \frac{\pi(k)}{k}}{\pi(n)}=1 \tag{1}$$

I call it "weak" because it seems less stringent than the prime number theorem: $\pi(x)\sim\frac{x}{\ln(x)}$

$$\sum_{k=1}^n \frac{\pi(k)}{k} \sim \int_1^n \frac{\pi(t)}{t}dt \ \overset{?}{\sim}\ \pi(n)$$

Therefore we obtain that the weak PNT is equivalent to: $$\pi'(x)\ \overset{?}{\sim}\ \frac{\pi(x)}{x} \Rightarrow \frac{1}{x}\ \overset{?}{\sim}\ \frac{\pi'(x)}{\pi(x)}\Rightarrow \ln(x)\ \overset{?}{\sim}\ \ln(\pi(x))=\ln(x)+\ln\left(\frac{\pi(x)}{x}\right)$$

$$\lim_{n\to \infty} \frac{\ln \left(\frac{\pi(x)}{x}\right)}{\ln(x)}=0 \tag{2}$$

$ (1) $ or $ (2) $ only require that $ \frac {\pi(x)} {x} \to 0 $ slower than any polynomial because if we had $\pi(x)\sim x^{1/c}\Rightarrow\frac{\ln(\pi(x)/x)}{\ln(x)}\sim c-1$

Let $ f (x) $ be an increasing function and $0< \alpha < 1$ such that $\lim_{x\to \infty} \frac{f(x)}{x^\alpha}=0$ $$O \left( \frac{x}{f(x)}\right)\leq\pi(x)\leq x \tag{3}$$

$(1)$, $(2)$ and $(3)$ are equivalent.

it is easy to see from $ (3) $ that this statement is much weaker than the prime number theorem. in fact it is trivial that $ \pi (x) <x $ therefore it is sufficient to show that

$N>1,N\neq\infty$ $$\pi(x)>O\left(\frac{x}{\ln(x)^N}\right) \tag{4}$$

As Chebyshev managed to prove by "elementary" means that $\pi(x)=O\left(\frac{x}{\ln(x)}\right)$ I was wondering how to prove $ (4) $ in an even more "elementary" way.

Patrick Danzi
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  • There is an "elementary" proof of the prime number theorem, so there is certainly a proof of the weak form you introduced. – TravorLZH Dec 21 '21 at 15:11
  • Chebyshev proved both inequalities, so $cx/\log x < \pi(x) < Cx/\log x$ and that trivially implies your result – Conrad Dec 21 '21 at 15:16
  • @Conrad considering Chebyshev's results it is trivial to prove my thesis. what I expect is that this theorem will be easier to prove than either the prime number theorem or the Chebychev estimate. So I am wondering if there is a proof of weak PNT, which uses a "simple math". – Patrick Danzi Dec 21 '21 at 15:53
  • @TravorLZH considering Chebyshev's results it is trivial to prove my thesis. what I expect is that this theorem will be easier to prove than either the prime number theorem or the Chebychev estimate. So I am wondering if there is a proof of weak PNT, which uses a "simple math". – Patrick Danzi Dec 21 '21 at 15:53
  • Chebyshev estimates (in a rough form with non optimal constants) are as simple math as it gets - see Hardy and Wright for a very simple proof of them – Conrad Dec 21 '21 at 16:01
  • In fact, one recovers Chebyshev function from the integral: $$\int_1^n{\pi(t)\over t}\mathrm dt=\pi(x)\log x-\vartheta(x)$$ – TravorLZH Dec 21 '21 at 16:20

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