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 result is true if 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/)".


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Later story(added just for additional information): The proof uses the "Deuring-Heilbronn phenomenon" of repulsion of exceptional zeros of a Dirichlet $L$-function, which is also rather interesting. This method was later strengthened by Landau, Siegel and so on, and finally with more recent developments on the Birch-Swinnerton-Dyer conjecture by Gross and Zagier, an effective version of this theorem was proved by Dorian Goldfeld, and the explicit constants were computed by Oesterlé. Thus the Gauss class number problem was solved in its entirety.