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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 there are no Siegel zeros of L-functions for imaginary quadratic fields. Siegel zeros are exceptional zeros occurring in the Generalized Riemann hypothesis is truereal line in the interval $(\frac{1}{2}, 1)$. The Deuring-Helbronn part(an exposition here) part is that the result is true if the Generalized Riemann hypothesis there are Siegel zeros. The proof uses an effect of "repulsion" of such zeros, which is falsecalled the Deuring-Heilbronn phenomenon. This is all explained by Dorian Goldfeld in a bulletin article, "Gauss' Class number problem for Imaginary Quadratic Fields".

The proof uses existence of Siegel zeros is a weaker version of the "Deuring-Heilbronn phenomenon" negation of repulsion the Generalized Riemann hypothesis. Hopefully in future the generalized Riemann hypothesis would be proved and thus hopefully it will be shown that the study of exceptional Sigel zeros had been just the study of a Dirichlet $L$-function, which is also rather interestingthe empty set.

Later story(added just for additional information): 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 Joseph Oesterlé. Thus the Gauss class number problem was solved in its entirety.

Thanks to Keith Conrad for correcting ambiguities.

4 added 396 characters in body

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) 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".

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(just saying)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.

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 proof uses the "Deuring-Heilbronn phenomenon" of repulsion of exceptional zeros of a Dirichlet $L$-function, which is also rather interesting(just saying).