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I like the proof in Newman's GTM book "Analytic number theory". He calls his presentation a natural (as opposed to "elementary" or "simple") proof. This is to say that the presentation emphasizes a motivated sequence of steps that illustrates how would might have discovered it. This proof (as well as many other "simple" and "elementary" proofs) proceeds using Landau's lemma on the abscissa of convergence of Dirichlet series with non-negative coefficients.