We consider the range of the random function [F(z)=∑^∞_n=0pm a_nz^n] (F chosen at random in the natural way) defined in D, where ∑|a_n|^2=∞. Is the image of w=F(z) with probability one [(a)] ; everywhere dense in the plane? ; the whole plane? ; does it contain any given point with probability one? If a_n=n^lambda, (b) holds if lambda>frac12, and (a) holds if -frac12<lambda<+frac12. (J. P. Kahane)

Research Problems in Function Theory — Problem 7.6

Kahane’s problem asks whether a random signed power series has an image that is almost surely dense in the plane, equal to the whole plane, or contains each prescribed point. Its stated power-law cases are known: density for $-\frac{1}{2}<\lambda<\frac{1}{2}$ and surjectivity for $\lambda>\frac{1}{2}$ .

Known results

For $a_n=n^\lambda$ , the image is almost surely dense when $-\frac{1}{2}<\lambda<\frac{1}{2}$ (Kahane).
For $a_n=n^\lambda$ , the image is almost surely the whole plane when $\lambda>\frac{1}{2}$ (Kahane).

Related random-analytic-function result

A recent preprint proves almost-sure density for broad classes with rotationally invariant complex coefficients, and whole-plane coverage for unbounded critical-regular Gaussian series; it also states a whole-plane result when $\sum a_n^2=\infty$ in the rotationally invariant setting. These coefficients are not the independent signs $\pm a_n$ in Problem 7.6, so the exact problem remains unsettled.

Current status (as of August 2026): The quoted power-law cases are settled, but the general independent-sign problem in Problem 7.6 has no recorded direct resolution; only analogous complex-coefficient results are available.

Research Problems in Function Theory — Problem 7.6

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