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By Kontodimopoulos N., Niakas D.

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3. 5. 4. 8. 5. If Ω is a C ∞ Jordan domain in u ∈ C ∞ (Ω) and for m = 0, 1, . . 18) holds. = 0, 1, . . 34). 6. Prove that Du = Licensed to AMS. 2/02 Chapter 13 Some Topics in Analytic Number Theory It was said that whoever proved the Prime Number Theorem would attain immortality. Sure enough, both Hadamard and de la Vall´ee Poussin lived into their late nineties. It may be that there is a corollary here. It may be that the Riemann Hypothesis is false: but, should anyone manage to actually prove its falsehood—to find a zero off the critical line—he will be struck dead on the spot, and his result will never become known.

6. 28), prove that (ϕ∗ πD )(z) = 1/ Im z. 31). 7. 34) (a) Prove that there is a strictly monotone function Ψ from (0, ∞) to z− (0, ∞) so that γ(z, w) = Ψ(ρπC+ (z, w)). ) (b) Prove for any Herglotz function, F , that γ(F (z), F (w)) ≤ γ(z, w). Remark. We put “metric” in quotes because the triangle inequality may not hold. However, the formula for γ is somewhat simpler than for ρ. 8. Let f be an isometry on a metric space X. Suppose f (x) = x. Prove that f [n] (x) cannot cannot have a convergent limit.

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