Abstract: In a seminal 1859 paper, Riemann gave two proofs of the analytic continuation and functional equation of his zeta function. The ideas behind his theta function proof were later developed into a powerful theory of Fourier analysis on number fields, in work of Hecke, Tate and others. In this talk, I will focus instead on the contour integral proof, and based on the ideas therein, will present two infinite families of new proofs of the analytic continuation and functional equation. The proofs are facilitated by geometric data coming from the fact that the polylogarithm generating function is a flat section of the universal unipotent bundle with connection over $\mathbb{P}^{1} \backslash \{0,1,\infty\}$.