If you are asking for an example of such a method in action, then you have the theorem of Hecke-Deuring-Heilbronn that $h(D) \rightarrow \infty$ as $D \rightarrow \infty$, where $h(D)$ is the class number of the imaginary quadratic field with discriminant $D$. The Hecke part is that the result is true if the Generalized Riemann hypothesis is true. Deuring-Helbronn part(an exposition [here](http://www.ams.org/notices/200608/fea%2Dstopple.pdf)) is that the Generalized Riemann hypothesis is false. This is all explained by Dorian Goldfeld in a bulletin article, "[Gauss' Class number problem for Imaginary Quadratic Fields](http://www.ams.org/journals/bull/1985-13-01/S0273-0979-1985-15352-2/)".