For 10 points, name this 19th century German mathematician, who defined his namesake integral as a limit of approximating sums.
This mathematician proved an inequality relating the dimension of the vector space of meromorphic functions on a one-dimensional complex manifold with the genus of the manifold, which was later strengthened by Roch. He names manifolds equipped with a positive definite symmetric bi-linear form g, the metric tensor.  The study of such manifolds constitutes his namesake geometry. In his sole paper on analytic number theory he defined an analytic continuation of the infinite p-series to the complex plane and conjectured that its nontrivial zeroes have real part one-half, later generalized to a conjecture about the zeroes of Dirichlet L-functions. The distribution of the prime numbers is governed by the zeroes of his namesake zeta function. For 10 points, name this 19th century German mathematician, who defined his namesake integral as a limit of approximating sums.
This mathematician proved an inequality relating the dimension of the vector space of meromorphic functions on a one-dimensional complex manifold with the genus of the manifold, which was later strengthened by Roch.
Bernhard {Riemann}