Nevanlinna Theory - Second Fundamental Theorem

Second Fundamental Theorem

We define N(r, f) in the same way as N(r,f) but without taking multiplicity into account (i.e. we only count the number of distinct poles). Then N₁(r,f) is defined as the Nevanlinna counting function of critical points of f, that is

The Second Fundamental theorem says that for every k distinct values a_j on the Riemann sphere, we have

This implies

where S(r,f) is a "small error term".

For functions meromorphic in the plane, S(r,f) = o(T(r,f)), outside a set of finite length i.e. the error term is small in comparison with the characteristic for "most" values of r. Much better estimates of the error term are known, but Andre Bloch conjectured and Hayman proved that one cannot dispose of an exceptional set.

This theorem is called the Second Fundamental Theorem of Nevanlinna Theory, and it allows to give an upper bound for the characteristic function in terms of N(r,a). For example, if f is a transcendental entire function, using the Second Fundamental theorem with k = 3 and a₃ = ∞, we obtain that f takes every value infinitely often, with at most two exceptions, proving Picard's Theorem.

As many other important theorems, the Second Main Theorem has several different proofs. The original proof of Nevanlinna was based on the so-called Lemma on the logarithmic derivative, which says that m(r,f'/f) = S(r,f). Similar proof also applies to many multi-dimensional generalizations. There are also differential-geometric proofs which relate it to the Gauss–Bonnet theorem. The Second Fundamental Theorem can also be derived from the metric-topological theory of Ahlfors, which can be considered as an extension of the Riemann–Hurwitz formula to the coverings of infinite degree.

The proofs of Nevanlinna and Ahlfors indicate that the constant 2 in the Second Fundamental Theorem is related to the Euler characteristic of the Riemann sphere. However, there is a very different explanations of this 2, based on a deep analogy with number theory discovered by Charles Osgood and Paul Vojta. According to this analogy, 2 is the exponent in the Thue–Siegel–Roth theorem. On this analogy with number theory we refer to the survey of Lang (1997) and the book by Min Ru (2001).