There are a number of formulas variously known as Hurwitz's formula.
The first is
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.