Original reference for Riemann's inequality Let $D$ be a divisor on a compact Riemann surface of genus $g$. The inequality
$$
l(D)\geq {\textrm {deg}}(D)-g+1
$$
is called Riemann's inequality.
$ \phantom{aaaaaaaa}$In which of Riemann's papers did this inequality first appeared?
 A: B. Riemann, Theorie der Abel'schen Functionen,  Journal für die reine und angewandte Mathematik 54, 101–155 (1857).
Here is a description of this contribution, by Jeremy Gray:

In this 1857 paper Riemann established the existence of complex
  functions on a surface with no boundary. He supposed the surface was
  p-fold connected  which means that it is rendered simply connected by p cuts when it forms a p-sided polygon.  He showed that there
  are p linearly independent everywhere holomorphic functions defined
  inside the polygon by considering what would happen if the real parts of their periods all vanished (using the Dirichlet principle). Later he showed that the differentials of these functions are
  everywhere defined holomorphic integrands.  Then he specified d
  points at which the function may have simple poles,  again imposing the
  condition that the functions jump by a constant along the cuts.  Now
  he argued that to create functions with only simple poles and constant
  jumps one should take a sum of  p linearly independent functions with no
  poles plus functions of the form $1/z$ at one of the specified points
  and add a constant term.  The resulting expression depends linearly
  on $p +  d+1$     constants.  The jumps therefore depend linearly on
  $p +  d+1$    constants  and there are  $p$ of them to be made to
  vanish (if the function is single valued as required).   So there will
  be non-constant meromorphic functions when $p +  d+1-2p\geq 2$, i.e.
  $d>p$. This result, today called the Riemann inequality,  says there
  is a linear space of complex functions of dimension $h^0>d+1-p$    and
  this contains non constant functions as soon as $d+1-p>1$ or $d>p$.

A: *

*Bernhard Riemann "Theorie der Abel'schen Functionen" (1857)


You can find a digital copy in here, pages 115 to 155.
