By Jan-Hendrik Evertse

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You may compare this with the function ordp for prime numbers p defined in Chapter 1. Both ordz0 and ordp are examples of discrete valuations. A discrete valuation on a field K is a surjective map v : K → Z ∪ {∞} such that v(0) = ∞; v(x) ∈ Z for x ∈ K, x = 0; v(xy) = v(x) + v(y) for x, y ∈ K; and v(x + y) min(v(x), v(y)) for x, y ∈ K. Let U ⊂ C be an open set. A set S called discrete in U in U if S ⊂ U and for every compact set K with K ⊂ U , the intersection S ∩ K is finite. By the Bolzano-Weierstrass Theorem, S is discrete in U if and only if S ⊂ U and S has no limit point in U , that is, there are no z0 ∈ U and an infinite sequence {zn } of distinct elements of S such that limn→∞ zn = z0 .

Let U be a non-empty, open, connected subset of C, and let f : U → C be an analytic function that is not identically 0 on U . , every compact subset of U contains only finitely many zeros of f . Proof. Suppose that some compact subset of U contains infinitely many zeros of f . Then by the Bolzano-Weierstrass Theorem, the set of these zeros would have a limit point in this compact set, implying that f = 0 on U . 20. Let U be a non-empty, open, connected subset of C, and f, g : U → C two analytic functions.

7. Let U ⊂ C be a non-empty, open, simply connected set, and f : U → C an analytic function. Then there exists an analytic function F : U → C with F = f . Further, F is determined uniquely up to addition with a constant. Proof (sketch). If F1 , F2 are any two analytic functions on U with F1 = F2 = f , then F1 − F2 is constant on U since U is connected. This shows that an anti-derivative of f is determined uniquely up to addition with a constant. It thus suffices to prove the existence of an analytic function F on U with F = f .

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Analytic Number Theory [lecture notes] by Jan-Hendrik Evertse
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