The Riemann zeta function was introduced by Euler, but carries Riemann's name because he was the one who extended it to a meromorphic function on the entire complex plane, and discovered its importance for the distribution of primes. It admits a vast class of generalizations, called L-functions, but, as in Riemann's case, one usually cannot prove anything about them without relying on seemingly unrelated integral representations.

 

In joint work with David Ben-Zvi and Akshay Venkatesh, we elucidate the origin of such integral representations, showing that they are manifestations of a duality between nice Hamiltonian spaces for a pair $(G,\check G)$ of ``Langlands dual'' groups. Over the geometric cousins of number fields -- algebraic curves and Riemann surfaces -- such dualities had been anticipated and constructed in many cases by Gaiotto and others, motivated by mathematical physics.