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

By Jan-Hendrik Evertse

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It is well-known that ζ(s) converges absolutely for all s ∈ C with Re s > 1, and that it diverges for s ∈ C with Re s 1. Moreover, ζ(s) defines an analytic (complex differentiable) function on {s ∈ C : −s Re s > 1}. Riemann obtained another expression for ∞ that can be defined n=1 n everywhere on C \ {1} and defines an analytic function on this set; in fact it can be −s shown that it is the only analytic function on C \ {1} that coincides with ∞ n=1 n on {s ∈ C : Re s > 1}. This analytic function is also denoted by ζ(s).

It follows that Lf (s) = limN →∞ Lf,N (s) converges if Re s > 0. 24 to the sequence of partial sums {Lf,N (s)}. Let K be a compact subset of {s ∈ C : Re s > 0}. There are σ > 0, A > 0 such that Re s σ, |s| A for s ∈ K.

Dt dx D n=0 54 So the conditions of the Fubini-Tonelli Theorem are satisfied, and in the expression for F (w) derived above we can interchange the integrations and the summation. 10, we obtain ∞ 1 (w − z)n F (w) = n=0 D 0 D 1 2πi ∞ (w − z)n = n=0 ∞ (w − z)n = D n=0 f (x, z + 2δe2πit ) dt dx (2δe2πit )n γz,2δ f (x, ζ) · dζ (ζ − z)n+1 dx f (n) (x, z) · dx . n! This shows that indeed, F (w) has a Taylor expansion around z converging on D(z, δ). So in particular, F is analytic in z. Further, F (k) (z) is equal to k!

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