This book provides a lucid exposition of the connections between
non-commutative geometry and the famous Riemann Hypothesis,
focusing on the theory of one-dimensional varieties over a finite
field. The reader will encounter many important aspects of the
theory, such as Bombieri's proof of the Riemann Hypothesis for
function fields, along with an explanation of the connections with
Nevanlinna theory and non-commutative geometry. The connection with
non-commutative geometry is given special attention, with a
complete determination of the Weil terms in the explicit formula
for the point counting function as a trace of a shift operator on
the additive space, and a discussion of how to obtain the explicit
formula from the action of the idele class group on the space of
adele classes. The exposition is accessible at the graduate level
and above, and provides a wealth of motivation for further research
in this area.
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