A question of Gol’dberg concerning entire functions with prescribed zeros
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Let (zj) be a sequence of complex numbers satisfying ¦zj¦→ ∞ asj → ∞ and denote by n(r) the number of zj satisfying ¦zj¦≤ r. Suppose that lim infr → ⇈ log n(r)/ logr > 0. Let ϕ be a positive, non-decreasing function satisfying ∫∞ (ϕ(t)t logt)−1 dt < ∞. It is proved that there exists an entire functionf whose zeros are the zj such that log log M(r,f) = o((log n(r))2ϕ(log n(r))) asr → ∞ outside some exceptional set of finite logarithmic measure, and that the integral condition on ϕ is best possible here. These results answer a question by A. A. Gol’dberg.
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