There are a number of formulas variously known as Hurwitz's formula.
The first is
![zeta(1-s,a)=(Gamma(s))/((2pi)^s)[e^(-piis/2)F(a,s)+e^(piis/2)F(-a,s)],](/images/equations/HurwitzsFormula/NumberedEquation1.svg) |
where 
is a Hurwitz zeta function, 
is the gamma function,
and 
is the periodic zeta function (Apostol 1995;
1997, p. 71).
Hurwitz has another formula, also known as Hurwitz's theorem or the Riemann-Hurwitz formula. Let 
and 
be compact Riemann
surfaces, and suppose that there is a non-constant analytic
map 
.
The Hurwitz formula gives the relationship between the genus
of 
and the genus of 
,
namely,
 |
In this formula, 
is the degree of the map. The degree of 
is an integer 
such that for a generic point 
, (i.e., for all but finitely many points in 
), the set 
consists of 
points in 
. The sum 
in the Hurwitz formula can be viewed as
a correction term to take into account the points where 
. Such points are sometimes called branch
points. The numbers 
are the ramification indices.
Hurwitz's theorem for Riemann surfaces essentially follows from an application of the polyhedral formula. It is used to find the genus of modular curves and hyperelliptic curves, and is often applied to find the genus of a complicated Riemann surface that happens to map to a simpler surface, usually the sphere.
