Clearing up proof of: Harmonic series of primes diverges

Question

In the book "An Introduction to Number Theory" by Apostol there is a proof given by Clarkson, that $ \sum_{n=1}^{\infty} {1 \over {p_m}} $ diverges.

It is assumed $ \sum_{m=k+1}^{\infty} {1 \over {p_m}} < {1 \over 2} $ for a certain $ k \in \mathbb{N} $ . Then it defined $ Q $ as: $ Q = \prod_{i=1}^{m} p_m $

And reaches the equation: $ \sum_{n=1}^{r} {1 \over 1+n*Q} \le \sum_{t=1}^{\infty} (\sum_{m=k+1}^{\infty} {1 \over p_m}) ^t $, which is perfectly fine.

Now it is claimed that the "right side includes among its terms all the terms of the left." This I don't understand.

Well, it is obvious for $ n = 1 $ that $ n*Q+1 $ is a new prime and thus included on the right side. However I don't see how the quoted statement above holds, when $1+Q*n$ is not prime.

Though I find it clear, that all prime factors of $1+n*Q$ appear in the sum of the right side.

I would be very happy, if someone could clear up this misunderstanding, as the problem seems rather basic and I still fail to get it right...

Answer 1

As has been pointed out, $Q+1$ need not be prime. But it's clear that none of $p_1,\dots,p_m$ can divide $1+n*Q$, since those primes all divide $Q$. So every $1+n*Q$ is a product of primes $p_j$ for $j>m$, and the reciprocal of any such product is one of the terms on the right.

Answer 2

For $n>0$ the number $nQ+1$ is a product of powers of primes that are all greater than $p_m$. The terms in the multinomial expansion of the RHS are reciprocals of products of powers of primes that are $>p_m$, and includes every such one at least once (Exactly once,actually,because prime decomposition is unique). So $1/(nQ+1)$ occurs as a RHS term for $ n=1...r$..... This is a cumbersome variant of Euler's proof : If $p_1,...,p_m$ are the only primes,then the set $S$ of all products of non-negative powers of $p_1,...,p_m $is $Z^+$. But $$\infty > M=\prod_{j=1}^{j=m} (1-1/p_j)^{-1} = \prod_{j=1}^{j=m} \sum_{k=0}^{k= \infty}p_j^{-k}$$ and multiplying out the RHS , the term $1/t$ occurs on thr RHS for every $t \in S$, so $$M \ge \sum_{t \in S} 1/t =\sum_{n=1}^{\infty}1/n=\infty$$ a contradiction.