The Riemann Hypothesis and its generalizations

The Riemann Hypothesis for $L(s)$ is the assertion that the nontrivial zeros of $L(s)$ lie on the critical line. For historical reasons there are names given to the Riemann hypothesis for various sets of $L$-functions. For example, the Generalized Riemann Hypothesis (GRH) is the Riemann Hypothesis for all Dirichlet $L$-functions. More examples collected below.

In certain applications there is a fundamental distinction between nontrivial zeros on the real axis and nontrivial zeros with a positive imaginary part. Here we use the adjective modified to indicate a Riemann Hypothesis except for the possibility of nontrivial zeros on the real axis. Thus, the Modified Generalized Riemann Hypothesis (MGRH) is the assertion that all nontrivial zeros of Dirichlet $L$-functions lie either on the critical line or on the real axis.

Nontrivial zeros which are very close to the point $s=1$ are called Landau-Siegel zeros.

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