FOM: Test Case for "Elementary" Proofs

[Walter Felscher]

Mr. Shipman wrote on December 11th about Dirichlet's theorem

> Dirichlet's theorem (If n,k>1 have no common factor then
> the sequence [n+k, > n+2k,n+3k,n+4k, ... ] contains a prime)
> is much simpler to state than the Prime > Number Theorem,
> and unlike the P.N.T. no proof has ever been found that does
> not go through complex analysis.

Already the titles of the basic articles

  Atle Selberg: An elementary proof of DIRICHLET's theorem
  about primes in an arithmetic progression. Ann.Math. 50
  (1949) 297-304 ,

  Atle Selberg: An elementary proof of the prime-number
  theorem for arithmetic progressions. Canad.J.Math. 2
  (1950) 66-78 ,

  Harold N.Shapiro: Some assertions equivalent to the prime
  number theorem for arithmetic progressions. Comm.pure
  appl.Math. 2 (1949) 293-308 ,

and much more so their contents, should make it clear that
Dirichlet's theorem has been proved there in the number
theorist's sense of 'elementary', namely without use of
complex analysis. [Of course, in contrast to the prime
number theorem, the notion of a 'character' is needed, i.e.
of (necessarily complex) roots of unity - but nothing about
complex analytic functions, their integration, the zeta
function &c. &c.: real analysis suffices.]

W.F.