In Tom Apostol's book, he credits the proof of the divergence of the sum of reciprocal of primes to James A. Clarkson.
Theorem: Let $\{p_n\}_{n\in\mathbb N}$ be the prime numbers. The infinite series $$\sum_{n=1}^\infty\frac{1}{p_n}$$ diverges.
Proof. We assume the series converges and obtain a contradiction. If the series converges there is an integer $k$ such that $$ \sum_{m=k+1}^{\infty} \frac{1}{p_m}<\frac{1}{2} . $$ Let $Q=p_1 \cdots p_k$, and consider the numbers $1+n Q$ for $n=1,2, \ldots.$ None of these is divisible by any of the primes $p_1, \ldots, p_k$. Therefore, all the prime factors of $1+n Q$ occur among the primes $p_{k+1}, p_{k+2}, \ldots.$ Therefore for each $r \geq 1$ we have $$ \sum_{n=1}^r \frac{1}{1+n Q} \leq \sum_{t=1}^{\infty}\left(\sum_{m=k+1}^{\infty} \frac{1}{p_m}\right)^t \tag{1} $$ since the sum on the right includes among its terms all the terms on the left. But the right-hand side of this inequality is dominated by the convergent geometric series $$ \sum_{t=1}^{\infty}\left(\frac{1}{2}\right)^t $$ Therefore the series $\sum_{n=1}^{\infty} 1 /(1+n Q)$ has bounded partial sums and hence converges. But this is a contradiction because the integral test or the limit comparison test shows that this series diverges.
My questions:
- I know that all the prime factors of $1+nQ$ must be a subset of $\{p_n\}_{n=k+1}^\infty$, but I don't see how every term on the left in $(1)$ appears on the right, can someone clarify this for me?
- I don't see how this leads to the infinitude of the primes. It seems we first must assume there are infinitely many in order to make this argument.
Source: Tom M. Apostol (1976) Introduction to Analytic Number Theory.
From the definition of $Q$ it follows that each $1+nQ$ can only be divisible by primes larger than $p_k$. Thus we can write each $$ \frac{1}{1+nQ} = \frac{1}{p_{i_1}p_{i_2}\cdots p_{i_M}} $$ for some $M=M(n)$ where $k+1\le i_1,i_2,\ldots,i_M$ (and the $i$ are not necessarily distinct, allowing primes to appear more than once in the factorization). Then this term must appear in the expansion of $$ \left(\frac{1}{p_{k+1}}+\frac{1}{p_{k+2}}+\frac{1}{p_{k+3}}+\cdots\right)^M $$
Although this is written with the implicit assumption of the infinitude of primes, it is not necessary for the argument. We can just write $$ \sum_{n=1}^r \frac{1}{1+nQ}\leq\sum_{t=1}^\infty\left(\sum_{n>k}\frac{1}{p_n}\right)^t $$ allowing the set of remaining primes to be finite or infinite, and the result is the same.
In Apostol the infinitude of primes is established before this result. I don't think he intends to say that it follows from this proof, he only mentions in a historical note that Euler proved this result in 1737 (presumably by a different method) and noted the implication.