I've been reading proofs of this theorem, and I was wondering why the following is true:

We know that $\sum\limits_{n=1}^{\infty}\frac{1}{n}$ diverges. I'm not sure how knowing this fact leads us to the next step:

For any $M >0$ we can find primes $p_1,\cdots p_m$ such that $\left(\sum\limits_{i=1}^{N}\frac{1}{p_1^i}\right)\cdots \left(\sum\limits_{i=1}^{N}\frac{1}{p_m^i}\right)>M.$

What I am saying is, given the opportunity, I would have never thought to look at a finite product of partial sums. I understand why everything works after the above step.

  • $\begingroup$ What is $ k $ ? $\endgroup$ – nbubis Feb 13 '14 at 6:10
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    $\begingroup$ Since $\sum \frac{1}{n}$ diverges, for some $N$ we have $\sum_1^N \frac{1}{n}\gt M$. Let the $p_i$ be all the primes less than $N$. Then your product is greater than $\sum_1^N \frac{1}{n}$. This is because when you expand you get all the $\frac{1}{n}$ with $n\le N$, plus some more terms. $\endgroup$ – André Nicolas Feb 13 '14 at 6:13
  • $\begingroup$ Why are we taking them to powers? $\endgroup$ – emka Feb 13 '14 at 6:13
  • $\begingroup$ Because some of the $\frac{1}{n}$ are products of powers of primes $\le$ the largest prime $\le N$. We use the unique factorization theorem. $\endgroup$ – André Nicolas Feb 13 '14 at 6:15
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    $\begingroup$ @Morris: I think it would benefit you to "flesh out the details" yourself as an exercise. $\endgroup$ – Bombyx mori Feb 13 '14 at 6:19

Note: This does not answer your question. I have changed the indices on your product. We start at $0$, not $1$. I believe that the altered indices are the "right" thing to use.

Let $M$ be any real number. Since $\sum \frac{1}{n}$ diverges, for some $N$ we have $\sum_1^N \frac{1}{n}\gt M$. Let the $p_1,p_2,\dots,p_m$ be all the primes $\le N$.

Now imagine expanding "your" (altered) product $$\left(\sum\limits_{i=0}^{N}\frac{1}{p_1^i}\right)\cdots \left(\sum\limits_{i=0}^{N}\frac{1}{p_m^i}\right).\tag{1}$$ We get all the $\frac{1}{k}$, where $k$ is a product of the form $$k=p_1^{a_1}p_2^{a_2}\cdots p_m^{a_m},$$ where the $a_i$ range independently over all numbers from $0$ to $N$, inclusive.

These $k$ include all integers in the interval from $1$ to $N$, plus possibly more. It follows that the product (1) is greater than or equal to (actually greater) than $\sum_1^N \frac{1}{n}$, and therefore (1) is greater than $M$.

  • $\begingroup$ Is this basically saying we are factoring $\sum\limits_{1}^{N} \frac{1}{k}$? $\endgroup$ – emka Feb 13 '14 at 7:19
  • $\begingroup$ Factoring each summand $1/k$. $\endgroup$ – Martín-Blas Pérez Pinilla Feb 13 '14 at 7:22
  • $\begingroup$ So something like $1+1/2+1/3+1/(2 \cdot 2)+1/5+1/(2 \cdot 3)+\cdots + 1/k$ $\endgroup$ – emka Feb 13 '14 at 7:23
  • $\begingroup$ Sort of, except the factorization is of $\sum_1^n \frac{1}{k}$ plus other terms beyond $\frac{1}{k}$. But these additional terms make the product bigger, which is good. $\endgroup$ – André Nicolas Feb 13 '14 at 7:23
  • $\begingroup$ Why would it include more? $\endgroup$ – emka Feb 13 '14 at 7:24

For $2\leq n\in \Bbb N$ let $S_n$ be the set of primes not exceeding $n.$ Let $x\in \Bbb N$ such that $2^x\geq n.$ $$\text {Let }\quad V(n)= \prod_{p\in S_n}\left(\frac {1}{1-\frac {1}{p}}\right).$$ $$\text {Then }\quad V(n) = \prod_{p\in S_n}\sum_{j=0}^{\infty}p^{-j}> \prod_{p\in S_n}\sum_{j=1}^xp^{-j}.$$ Completely expand the right-most expression above. For every $m\in \Bbb N$ with $2\leq m\leq n,$ the term $\frac {1}{m}$ will appear in the expansion as the reciprocal of a product of powers of some of the members of $S_n.$ So we have $V(n)>1+\sum_{m=2}^{n+1}\frac {1}{m}.$ Therefore $\lim_{n\to \infty}V(n)=\infty.$

Let $\pi$ be the set of all primes. Then $$\prod_{p\in \pi}(1-\frac {1}{p})=\lim_{n\to \infty}\prod_{p\in S_n}(1-\frac {1}{p})=\lim_{n\to \infty}V(n)^{-1}=0. $$

Theorem. (Elementary). If $0\leq a_j<1$ for all $j\in \Bbb N$ then $$\prod_{j=1}^{\infty}(1-a_j)=0\iff \sum_{j=1}^{\infty}a_j=\infty.$$

We have $\prod_{p\in \pi}(1-\frac {1}{p})=0.$ By the above theorem we have $\sum_{p\in \pi}\frac {1}{p}=\infty.$

This is due to Leonhard Euler.


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