(quoted from wikipedia)

The issue of "depth"
In the first half of the twentieth century, some mathematicians felt that there exists a hierarchy of techniques in mathematics, and that the prime number theorem is a "deep" theorem, whose proof requires complex analysis. Methods with only real variables were supposed to be inadequate. G. H. Hardy was one notable member of this group.

The formulation of this belief was somewhat shaken by a proof of the prime number theorem based on Wiener's tauberian theorem, though this could be circumvented by awarding Wiener's theorem "depth" itself equivalent to the complex methods. However, Paul Erdõs and Atle Selberg found a so-called "elementary" proof of the prime number theorem in 1949, which uses only number-theoretic means. The Selberg-Erdõs work effectively laid rest to the whole concept of "depth", showing that technically "elementary" methods (in other words combinatorics) were sharper than previously expected. Subsequent development of sieve methods showed they had a definite role in prime number theory.

Avigad et al. (2005) contains a computer verified version of this elementary proof in the Isabelle theorem prover.