From the discussion A simple way to obtain on math.stackexchange, I learnt an elementary proof of Euler product formula; in fact, I found it so remarkable that I decided to write a post on it.

We only need the two following basic lemmas; only sketchs of proof are given, we refer to Jean Jacod’s book, *Probability essentials*, for more information.

**Definition:** is a family of independent events if for every finite subset ,

.

**Lemma:** Let be a probability measure and be a family of independent events. Then

.

**Sketch of proof.** is a decreasing sequence of events, hence

.

**Lemma:** If is a familiy of independent events, then so is.

**Sketch of proof.** Using inclusion-exclusion principle, prove by induction on that every subfamily of cardinality satisfies

.

For , we have

**Theorem:** Let . Then .

**Proof.** Let be a random variable and be a probability measure such that

;

for example, take and . Let be the event “ is divisible by “. Notice that

.

Therefore, for any primes ,

.

We deduce that is a family of independent events. Because if and only if is not divisible by any prime, we have

.

### Like this:

Like Loading...

*Related*