1
Using calculus to do number theory
Calculus is the mathematics of approximations of continuous quantities ; number theory asks exact problems about discrete quantities. These seem, therefore, to be unrelated studies.
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2
What does the Riemann zeta function have to do with the distribution of primes?
One of the oldest theorems in mathematics is that there are infinitely many primes. Euclid (the geometer!) proved this around 300 BCE, and for centuries afterwards the story ended there. There are infinitely many primes -- so what else is there to ask about them?