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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 real line in the interval $(\frac{1}{2}, 1)$. The Deuring-Helbronn part is that the result is true if there are Siegel zeros. The proof uses an effect of "repulsion" of such zeros, which is called the Deuring-Heilbronn phenomenon. This is all explained by Dorian Goldfeld in a bulletin article, "Gauss' Class number problem for Imaginary Quadratic Fields".

The existence of Siegel zeros is a weaker version of the negation of 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 Sigel zeros had been just the study of the 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.

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 real line in the interval $(\frac{1}{2}, 1)$. The Deuring-Helbronn part is that the result is true if there are Siegel zeros. The proof uses an effect of "repulsion" of such zeros, which is called the Deuring-Heilbronn phenomenon. This is all explained by Dorian Goldfeld in a bulletin article, "Gauss' Class number problem for Imaginary Quadratic Fields".

The existence of Siegel zeros is a stronger version of the negation of 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 Sigel zeros had been just the study of the 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.

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 real line in the interval $(\frac{1}{2}, 1)$. The Deuring-Helbronn part is that the result is true if there are Siegel zeros. The proof uses an effect of "repulsion" of such zeros, which is called the Deuring-Heilbronn phenomenon. This is all explained by Dorian Goldfeld in a bulletin article, "Gauss' Class number problem for Imaginary Quadratic Fields".

The existence of Siegel zeros is a stronger version of the negation of 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 Sigel zeros had been just the study of the 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.

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 real line in the interval $(\frac{1}{2}, 1)$. The Deuring-Helbronn part is that the result is true if there are Siegel zeros. The proof uses an effect of "repulsion" of such zeros, which is called the Deuring-Heilbronn phenomenon. This is all explained by Dorian Goldfeld in a bulletin article, "Gauss' Class number problem for Imaginary Quadratic Fields".

The existence of Siegel zeros is a weaker version of the negation of 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 Sigel zeros had been just the study of the 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.

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 real line in the interval $(\frac{1}{2}, 1)$. The Deuring-Helbronn part is that the result is true if there are Siegel zeros. The proof uses an effect of "repulsion" of such zeros, which is called the Deuring-Heilbronn phenomenon. This is all explained by Dorian Goldfeld in a bulletin article, "Gauss' Class number problem for Imaginary Quadratic Fields".

The existence of Siegel zeros is a weaker version of the negation of 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 Sigel zeros had been just the study of the 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.

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 real line in the interval $(\frac{1}{2}, 1)$. The Deuring-Helbronn part is that the result is true if there are Siegel zeros. The proof uses an effect of "repulsion" of such zeros, which is called the Deuring-Heilbronn phenomenon. This is all explained by Dorian Goldfeld in a bulletin article, "Gauss' Class number problem for Imaginary Quadratic Fields".

The existence of Siegel zeros is a stronger version of the negation of 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 Sigel zeros had been just the study of the 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.