New Directions in Modularity
The problem of understanding prime numbers goes back to antiquity. One natural problem which arose in the 19th century was to count the number of primes less than X for some variable quantity X. The resolution came in an unexpected way : this *discrete* counting problem turned out to be related to the properties of a *continuous*  function first studied by Euler but now known as the Riemann zeta function. The breakthrough hinged on a crucial property of this function discovered by Riemann, namely, that it had a hidden symmetry which related the value of the function evaluated at the point "s" 