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?**



Original reference for Riemann's inequality
B. Riemann, [Theorie der Abel'schen Functionen](http://www.maths.tcd.ie/pub/HistMath/People/Riemann/AbelFn/), Journal für die reine und angewandte Mathematik **54**, 101–155 (1857).


Here is a description of this contribution, by [Jeremy Gray](https://eudml.org/doc/223239):



> 
> 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 $d>p$.
> 
> 
> 


