The sum of the reciprocals of the primes $p$ which are congruent to 1 mod 4 diverges to infinity, as does the sum over primes congruent to 3 mod 4. For example, Wikipedia claims the latter is a consequence of the strong form of Dirichlet's theorem on arithmetic question.

I was wondering if there are known elementary proofs of these results, i.e., without using the machinery of analytic number theory. I tried searching the published literature but found it very difficult to search for this sort of thing.


closed as off-topic by Jeremy Rouse, Will Sawin, Alexey Ustinov, Lucia, Mikhail Katz Jun 16 '17 at 7:13

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  • $\begingroup$ This may be a duplicate of mathoverflow.net/questions/16735/… , which appears in the "Related" sidebar. $\endgroup$ – LSpice Jun 15 '17 at 20:13
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    $\begingroup$ Dirichlet's original argument is really quite elementary. The special case that you ask about is done in Ireland and Rosen's "Classical Introduction to Modern Number Theory" in 4 pages (bottom of 249-top of 253). $\endgroup$ – Jeremy Rouse Jun 15 '17 at 20:25
  • $\begingroup$ @JeremyRouse Thanks for the helpful suggestion. I was unaware of that proof, and it certainly is simpler than other approaches I have seen, but I was most interested in a proof that does not use analytic number theory techniques. $\endgroup$ – pente Jun 15 '17 at 20:37
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    $\begingroup$ How do you prove an infinite series diverges without using analysis? if the terms of the series are defined by primes, is not that analysis then analytic number theory by definition? $\endgroup$ – Stopple Jun 15 '17 at 21:48
  • $\begingroup$ Erdos's proof for the sum of the reciprocals of all primes would be an example of an elementary proof. The proof mentioned by Jeremy uses Dirichlet L-series and zeta functions and convergence properties as s -> 1. $\endgroup$ – pente Jun 15 '17 at 22:10