The Second Main Theorem via Negative Curvature
Shortly after R. Nevanlinna’s first proof of the Second Main Theorem, Nevanlinna’s brother, F. Nevanlinna, gave a “geometric” proof of the Second Main Theorem. In this chapter, we give a geometric proof of the Second Main Theorem based on “negative curvat
- PDF / 2,624,019 Bytes
- 40 Pages / 439.37 x 666.142 pts Page_size
- 33 Downloads / 211 Views
Shortly after R. Nevanlinna's first proof of the Second Main Theorem, Nevanlinna's brother, F. Nevanlinna, gave a "geometric" proof of the Second Main Theorem. In this chapter, we give a geometric proof of the Second Main Theorem based on "negative curvature," which broadly speaking, has the same overall structure as F. Nevanlinna's proof, although in terms of details, the proof we present here bears much greater resemblance to the work of Ahlfors [Ahlf 1941]. In Chapter 4, we will give another proof of the Second Main Theorem that is closer to R. Nevanlinna's original proof. Of course, neither the Nevanlinna brothers nor Ahlfors were interested in the exact structure of the error term. The error term we will present here is essentially due to P.-M. Wong [Wong 1989]. We have decided to begin with a negative curvature proof because obtaining a good error term using the geometric method requires less preparation than does the logarithmic derivative approach used in R. Nevanlinna's original proof. The geometric proof also has the virtue that the explicit constants occurring in the error terms have natural geometric interpretations. Another advantage of the geometric method is that one can use it to obtain the best possible error term for what's known as the "Ramification theorem," something which has yet to be accomplished using logarithmic derivatives. Moreover, in higher dimensions, the geometric approach has been successfully applied to a greater variety of situations than has the logarithmic derivative approach. On the downside, the structure of the error term here initially appears to be somewhat more complicated than the error term we will obtain in Chapter 4.
2.1 Khinchin Functions and Exceptional Sets Before proceeding directly to the Second Main Theorem, we introduce some notation and terminology that will help us in our discussion of "error terms." We begin with a definition. A function 1jJ will be called a Khinchin function if 1jJ is a real-valued, continuous, nondecreasing, and ~ 1 on the interval [e, 00), and 1jJ satisfies the following convergence condition (known as the Khinchin convergence condition): r~ 1 x1jJ(x) dx < 00.
le
W. Cherry et al., Nevanlinna’s Theory of Value Distribution © Springer-Verlag Berlin Heidelberg 2001
50
Second Main Theorem via Negative Curvature
[Ch.2]
In fact, since we will be interested in the value of the above integral, we will give it a name. Given a Khinchin function 'ljJ, define
ko
to
= ko('ljJ) = Je
1 x'ljJ(x) dx
< 00.
We will explain in §6.2 why we have chosen the names "Khinchin function" and "Khinchin convergence condition." We note that in the original work of Nevanlinna, 'ljJ (x) was taken to be (log x) He for positive c. One might find it helpful to keep this example in mind. Note that our choice of e as the start of the interval of definition of 'ljJ could be replaced with an arbitrary positive number, but because we have the prototypical example of 'ljJ (x) = (log x) 1+e in mind, and because we want 'ljJ (x) ~ 1, the number e is a convenient choice
Data Loading...
