Euler's proof of infinite primes and his product formula.

[Yuqing]

From what I had read, Euler had originally proved the infinitude of primes through proving his product formula and equating the primorial to the divergent harmonic series.

$$\sum^{\infty}_{n=1}\frac{1}{n^{s}} = \prod^{\infty}_{p}\frac{1}{1-p^{-s}}$$

where p ranges through the primes. However, the proofs of this formula that I have seen all seem to rely on the fact that the sum on the left is absolutely convergent for s>1. I am sure proofs of this formula can be easily found on the web so I am not going to reproduce them here, but I found the proof which employs the sieve of Eratosthenes quite fascinating. However, that proof too relies on rearrangement and at the last step, it requires that the

$$lim_{m\rightarrow\infty}\sum^{\infty}_{m}\frac{1}{n^{s}} = 0$$

which isn't the case for the harmonic series. So is the proof actually valid for the case s=1 (or s<1) by modern standards of rigor? If not, can somebody point me towards a valid proof? I assume that the result is true even if the proofs I found were invalid as it appears to be quite celebrated.

 

[mathman]

For your last equation (m->∞), the sum is divergent for all s ≤ 1 and all m, so the limit is obviously not 0.

 

[gb7nash]

If not, can somebody point me towards a valid proof?

Here's a proof that I like:

Suppose that there are a finite number of primes. So let the primes be p₁, p₂, ... , p_n.

Now consider the integer p₁p₂...p_n + 1. We know any integer can be decomposed into a unique factorization of primes. So:

p₁p₂...p_n + 1 = p₁^e₁p₂^e₂...p_n^e_n

where e_i <- N, where N includes 0. There exists a prime p_i such that e_i > 0. So rewriting the equation:

p_i(p₁p₂...p_i-1p_i+1...p_n - p₁^e₁p₂^e₂...p_i-1^e_i-1p_i^e_i-1p_i+1^e_i+1...p_n^e_n) = 1, so p_i divides 1, where we get the contradiction.

This is a huge abuse of notation, as i could be 1 or n. But you should get the idea.

 

[Yuqing]

mathman: I know, which is why is questioned the validity of the theorem.

gb7nash: I was actually looking for a proof of the Euler Product Formula, not an infinitude of primes. Thanks nevertheless, I should've been more clear.

I have read sources which state the theorem for only s > 1, and perhaps the proof and theorem does not hold for s = 1 or s < 1. So does that mean Euler's result cannot be regarded as true?

 

[HallsofIvy]

It was not Euler who gave that proof of the infinitude of primes, It goes back, at least, to Euclid.

 

[mathman]

Try google "Euler's product formula". I learned 2 things (1) the expression holds for Re(s) > 1 and (2) the divergence of the series for s = 1 implies that the number of primes is infinite, i.e. the right hand side product is infinite, (so the number of primes is infinite).

 

[Yuqing]

Try google "Euler's product formula". I learned 2 things (1) the expression holds for Re(s) > 1 and (2) the divergence of the series for s = 1 implies that the number of primes is infinite, i.e. the right hand side product is infinite, (so the number of primes is infinite).

This is sort of what I am asking, please tell me if I am misunderstanding or missing something.

1. The infinite primes are shown through the equality of the product to the series, since the series diverges for s = 1, then the product, being equal, also diverges (hence infinite).

2. If the formula holds for Re(s)>1 then why does the series for s = 1 imply that primes are infinite? The formula has not been shown to hold for s = 1 so how can we assume the equality?

 

[disregardthat]

I agree with you, the proof of the equality will not hold when s = 1. But what about this: for s = 2 the zeta-function yields pi^2/6 which is irrational. However, if there were a finite number of primes, the product formula would be rational. Another way is to look at the limit as s approaches 1. The zeta-function is continuous, so it will approach infinity. But the product formula would not if there were a finite number of primes.

 

[Yuqing]

Thank you Jarle, that seems quite satisfactory to me.

 

[camilus]

heres a link to the proof that i know.

http://a3.sphotos.ak.fbcdn.net/hphotos-ak-snc4/34831_1637819978449_1025224257_31748444_4847470_n.jpg