Infinitude of Primes Via Euler's Product Formula

In 1737, L. Euler published a paper where - among other things - he derived a formula that pointed to a wonderful connection between the harmonic series and prime numbers:

$\displaystyle 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}\ldots=\frac{2\cdot 3\cdot 5\cdot 7\cdot 11\cdot\ldots}{1\cdot 2\cdot 4\cdot 6\cdot 10\cdot\ldots}$

that could be written in modern concise notations as

(*)

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}=\prod_{p}\frac{p}{p-1}=\prod_{p}\frac{1}{1-1/p},$

where the product ranges over all primes.

Strictly speaking, the formula (and the way it was derived [Dunham, 65-70]) does not make sense as the harmonic series on the left is divergent. It did make sense to Euler, though. In 1876, Leopold Kronecker proved

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n^s}=\prod_{p}\frac{1}{1-p^{-s}},$

$s\gt 1,$ and then interpreted Euler's formulas is the outcome of passing to the right-sided limit as $s\rightarrow 1^{+}.$

Right or wrong, Euler's formula (*) has an imminently intuitive appeal. Each factor in the product on the right is the sum of a geometric series:

$\displaystyle 1+\bigg(\frac{1}{p}\bigg)+\bigg(\frac{1}{p}\bigg)^2+\bigg(\frac{1}{p}\bigg)^3+\ldots=\frac{1}{1-1/p},$

and, thus the formula can be rewritten as,

$\displaystyle\sum_{n=1}^{\infty}\frac{1}{n}=\prod_{p}\sum_{k=0}^{\infty}\frac{1}{p^k}=\prod_{p}(1+\frac{1}{p}+\frac{1}{p^2}+\frac{1}{p^3}+\ldots).$

To carry out the multiplication on the right, we need to pick up exactly one term from every sum that is a factor in the product and, since every integer admits a unique prime factorization, the reciprocal of every integer will be obtained in this manner - each exactly once.

We may exploit this idea to prove the infinitude of prime. Indeed, assume to the contrary that the number of primes is finite. Then the product on the right is finite. But the product expands into the harmonic series - the series of the reciprocals of all integers, which then, too, ought to be finite, but we know that it is not - a contradiction.

Reference

W. Dunham, Euler: The Master of Us All, MAA, 1999

Infinitude of Primes

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