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

By Jan-Hendrik Evertse

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Example. Let U ⊂ C be a non-empty, open, simply connected subset of C with 0 ∈ U . Then 1/z has an anti-derivative on U . For instance, if U = C \ {z ∈ C : Re z 0} we may take as anti-derivative of 1/z, Log z := log |z| + iArg z, 40 where Arg z is the argument of z in the interval (−π, π) (this is called the principal value of the logarithm). 3 (z − 1)n . 8. Let U ⊆ C be a non-empty, open set and f : U → C an analytic function. Further, let z0 ∈ U and R > 0 be such that D(z0 , R) ⊆ U . Then f has a Taylor series expansion ∞ an (z − z0 )n converging for z ∈ D(z0 , R).

After Hadamard and de la Vall´ee Poussin, several other proofs of the Prime Number theorem were given, all based on complex analysis. In the 1930’s, Wiener and Ikehara proved a general so-called Tauberian theorem (from functional analysis) which implies the Prime Number Theorem in a very simple manner. In 1948, Erd˝os and Selberg independently found an elementary proof, “elementary” meaning that the proof avoids complex analysis or functional analysis, but definitely not that the proof is easy!

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 . 39 Fix z0 ∈ U . Given z ∈ U , we define F (z) by f (w)dw, F (z) := γz where γz is any path in U from z0 to z. This does not depend on the choice of γz . For let γ1 , γ2 be any two paths in U from z0 to z.

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