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<h1 align=center> The Riemann Hypothesis and its generalizations
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The Riemann Hypothesis for $L(s)$ is the assertion that the
\begin{rawhtml}<a href="../64a">nontrivial zeros</a>\end{rawhtml} of $L(s)$ lie on the
\begin{rawhtml}<a href="../64a">critical line</a>\end{rawhtml}.  For historical reasons there are
names given to the Riemann hypothesis for various sets
of $L$-functions.  For example, 
the \begin{rawhtml}<a href="../17a">Generalized Riemann Hypothesis (GRH)</a>\end{rawhtml} is
the Riemann Hypothesis for all \begin{rawhtml}<a href="../9a">Dirichlet $L$-functions</a>\end{rawhtml}.
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 {\it modified} to indicate
a Riemann Hypothesis except for the possibility
of nontrivial zeros on the real axis.  Thus, the
\begin{rawhtml}<a href="../17a">Modified Generalized Riemann Hypothesis (MGRH)</a>\end{rawhtml} is
the assertion that all nontrivial zeros of 
\begin{rawhtml}<a href="../9a">Dirichlet $L$-functions</a>\end{rawhtml} lie either on the 
critical line or on the real axis.

Nontrivial zeros which are very close to the point $s=1$ are
called \begin{rawhtml}<a href="../36a">Landau-Siegel zeros</a>\end{rawhtml}.  

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Back to the 
<a href="http://www.aimath.org/WWN/rh/"> main index</a> 
for The Riemann Hypothesis.
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