Elementary Methods in Number Theory Exercise 1.5.15

Let $p_1,p_2,\cdots,p_k$ be a finite set of prime numbers.Prove that the number of positive integers $n\leq x$ that can be written in the form $n=p_1^{r_1}\cdots p_k^{r_k}$ is at most
\begin{equation}
\prod _{i=1}^k(\log_{p_i}x +1)
\end{equation}

Prove that if $x$ is sufficiently large,then there are positiveintegers $n\leq x$ that can not be represented in this way.Use this to give another proof that the number of primes is infinite.

Proof:

(1)Simple.

(2)I need to prove that
\begin{equation}
\lim_{x\to\infty}\frac{\prod _{i=1}^k(\log_{p_i}x+1)}{x}=0
\end{equation}
I just need to prove that
\begin{equation}
\lim_{x\to\infty}\prod_{i=1}^k \frac{\log_{p_i}x+1}{\sqrt[k]{x}}=0
\end{equation}

I just need to prove that $\forall 1\leq i\leq k$,
\begin{equation}
\lim_{x\to\infty}\frac{\log_{p_i}x}{\sqrt[k]{x}}=0
\end{equation}
This is obvious(Why?)

注:这道题给我很大启发:其实，素数有无限个是很容易从直观上理解的.假如只有有限个素数$p_1,p_2,\cdots,p_n$,那么根据算术基本定理，所有的正整数都可以写成这种形式:$$p_1^{r_1}p_2^{r_2}\cdots p_n^{r_n}$$这种形式无疑无法将每一个正整数表达出来，因为仔细观察会发现,$\forall 1\leq i\leq n$,$r_i$只要加上1，整个式子将翻好几倍，根本无法一格一格地变化.这种直观上的理解严格化，就体现在这篇博文中——它反映出素数之所以有无限个的原因是对数函数增长太慢，根本比不过幂函数的增长速度.或者换一句话说，是指数函数增长太快.