I was going through the paper 'Euclid’S theorem on the infinitude of primes: A historical survey of its proofs' by Romeo Mestrovic where he mentioned that L. Gegenbauer proved Infinitude of Primes by means of the series $\sum_{n=1}^{\infty} \frac{1}{ n^s}$,(p-$20$) which is the claim of Dickson taken from his book 'History of the theory of numbers, volume I, Divisibility and Primality ' (p-$413$). They both referred to the following paper which I am unable to find from internet.
L. Gegenbauer, Note ¨uber die Anzahl der Primzahlen, Sitzungsber, SBer. Kais. Akad. Wissensch. Wien (Math.) 95, II (1887), 94–96; 97, Abt.IIa (1888), 374–377.
I asked this question in the following link but the paper attached there is a paper of of Riemann,not of Gegenbauer.
I will be highly grateful if someone explain the proof mentioned in this paper or at least mention the source where I can find it. Thanks in advance
Suppose that there are only finitely many primes. Then the Euler product in $$ \frac{\pi^2}{6}=\sum_{n=1}^{\infty}\frac{1}{n^2}=\zeta(2)=\prod_{p\in \Bbb P}\frac{p^2}{p^2-1} $$ is rational, so that $\pi^2$ is rational. This is a contradiction.