principal divisor. Two divisors that differ by a principal divisor are called linearly equivalent. The divisor of a meromorphic 1-form is defined similarly. A divisor of a global meromorphic 1-form is called the canonical divisor (usually denoted ). Any two meromorphic 1-forms will yield linearly equivalent divisors, so the canonical divisor is uniquely determined up to linear equivalence (hence "the" canonical divisor). The symbol denotes the degree (occasionally also called index) of the divisor , i.e. the sum of the coefficients occurring in . It can be shown that the divisor of a global meromorphic function always has degree 0, so the degree of a divisor depends only on its linear equivalence class. The number is the quantity that is of primary interest: the dimension (over ) of the vector space of meromorphic functions on the surface, such that all the coefficients of are non-negative. Intuitively, we can think of this as being all meromorphic functions whose poles at every point are no worse than the corresponding coefficient in ; if the coefficient in at is negative, then we require that has a zero of at least that multiplicity at – if the coefficient in is positive, can have a pole of at most that order. The vector spaces for linearly equivalent divisors are naturally isomorphic through multiplication with the global meromorphic function (which is well-defined up to a scalar). Statement of the theorem The Riemann–Roch theorem for a compact Riemann surface of genus with canonical divisor states Typically, the number is the one of interest, while is thought of as a correction term (also called index of speciality) so the theorem may be roughly paraphrased by saying dimension − correction = degree − genus + 1. Because it is the dimension of a vector space, the correction term is always non-negative, so that This is called Riemann's inequality. Roch's part of the statement is the description of the possible difference between the sides of the inequality. On a general Riemann surface of genus , has degree , independently of the meromorphic form chosen to represent the divisor. This follows from putting in the theorem. In particular, as long as has degree at least , the correction term is 0, so that The theorem will now be illustrated for surfaces of low genus. There are also a number other closely related theorems: an equivalent formulation of this theorem using line