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Below is a well known Corollary from Analytic number theory and a proof (excerpt) by G. Fousserau (1892) which I have found here: Narkiewicz. (2000). The Development of Prime Number Theory on page 13.

Corollary 2. Let $\pi(x)$ be the prime counting function. Then $\pi(x)=o(x)$ applies for $x\rightarrow\infty$, i.e. $$ \lim _{x \rightarrow \infty} \frac{\pi(x)}{x}=0 . $$

Proof. For any $k>1$ we have $$ \pi(x) \leq k+\sum_{\substack{n \leq x \\\operatorname{gcd}(n, k)=1}} 1, $$ hence writing $x=q k+r$ with $0 \leq r<k$ we get $$ \pi(x) \leq k+q \varphi(k)+r \leq 2 k+\color{red}{\left\lfloor \frac{x}{k} \right\rfloor} \varphi(k) . $$

This implies $$ \limsup _{x \rightarrow \infty} \frac{\pi(x)}{x} \leq \frac{\varphi(k)}{k} . $$ [...]

I do not understand the estimate of $q$ by the $\color{red}{\text{floor function}}$. I get that $q\leq\frac{x}{k}$. This also does the job. Also, I think that it should be $<$ as $r\leq k-1$. Maybe someone can clarify some of the details here.

Note: The book says after the proof: See Mamangakis 1962.

I think this leads to this source here. In Fousseraus work I cannot really find the proof I am refering to. The proof in Mamangakis is a little bit different from the one in Narkiewicz.

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    $\begingroup$ Asking four questions in one does not fit well with the stackexchange model. What if four different users each answer one of the questions? Which one do you accept then? $\endgroup$
    – Gerry Myerson
    Commented Jun 20 at 11:54
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    $\begingroup$ I found this citation on a Japanese website: Fousserau, G. (1892): Sur la fr`equence des nombres premiers. Ann. Sci. Ec. Norm. Sup., (3), 9, 31–34. The paper is available at numdam.org/article/ASENS_1892_3_9__31_0.pdf $\endgroup$
    – Gerry Myerson
    Commented Jun 20 at 22:33
  • $\begingroup$ We can be quite sure that the surname is Fousserau, the name is harder to find. $\endgroup$
    – Sassatelli Giulio
    Commented Jun 21 at 9:19
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    $\begingroup$ The point about floor function is that $q=\lfloor x/k\rfloor$, and the inequality $\le$ is a consequence of applying $r<k$. $\endgroup$
    – TravorLZH
    Commented Jun 22 at 13:51
  • $\begingroup$ @TravorLZH But why is $q=\lfloor x/k\rfloor$. $\endgroup$
    – calculatormathematical
    Commented Jun 22 at 15:05

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