# Tag Info

### Simplest of the many proofs the prime harmonic series diverges

This is my favorite way to present Euler's proof: By the fundamental theorem of arithmetic: $$\prod_{p \leq N} \left(1+\frac{1}{p}+\frac{1}{p^{2}}+\cdots\right) \geq 1+\frac{1}{2}+ \cdots+\frac{1}{N}$$...
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### Simplest of the many proofs the prime harmonic series diverges

The simplest proof is Euler's original proof. It is based on the identity $$\sum_{n=1}^\infty n^{-s}=\prod_{p}\left(1-p^{-s}\right)^{-1}, s>1.$$ This identity is equivalent to existence and ...
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