In the first part of this talk I intend to do a brief survey on
the Riemann Hypothesis, one of the most important open problems of our time
(and worth 1,000,000 USD). On the second part of the talk I will show how
some extremal problems in Fourier analysis come into play when bounding
some objects related to the Riemann zeta-function, under the assumption of
the Riemann hypothesis. Most of the talk should be accessible to students
with a basic knowledge of complex analysis.