Hello everybody.
I'm looking for an "easy" example of a (nonzero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1} f(re^{i\phi})=0$.
Does anyone know such an example.
Best CJ
Hello everybody. I'm looking for an "easy" example of a (nonzero) holomorphic function $f$ with almost everywhere vanishing radial boundary limits: $\lim\limits_{r \rightarrow 1} f(re^{i\phi})=0$. Does anyone know such an example. Best CJ 


I am not sure if this would qualify as 'easy' but the first example of such a function was constructed by Lusin. It can be found in N. Lusin, J. Priwaloff, Sur l'unicité et la multiplicité des fonctions analytiques, Ann. Sci. École Norm. Sup. (3), 1925, p. 143191 (see p. 185). 


According to a footnote in the famous HardyRamanujan paper "Asymptotic formulae in combinatory analysis", the function $f(q)=\prod_{n=1}^{\infty}\frac{1}{1q^n}$ vanishes like $f(re^{i\theta})=o((1r)^{1/4\varepsilon})$ for almost all $\theta$. No proof is given, though I can't imagine Hardy would have made a statement like this without a proof in his pocket. Edit: This isn't actually hard to guess at. By Euler's pengatonal number theorem, we have $f(q)^{1}=\sum_{n\in \mathbf{Z}}(1)^{n}q^{n(3n1)/2}$, so Plancherel gives $\int_{0}^{2\pi}f(re^{i\theta})^{2}d\theta=2\pi\sum_{n\in \mathbf{Z}}r^{n(3n1)} \sim 2 \pi^{3/2}3^{1/2}(1r)^{1/2}.$ 


To complement Andrey Rekalo's response, Lusin's construction was generalized by Bagemihl and Seidel (Math. Zeitschrift 61 (1954), online here). See their Corollary 4 whose proof takes about 2 pages, much less than the original one by LusinPriwaloff. Of course the proof relies on Mergelyan's famous approximation theorem for which see Section 20.5 in Rudin: Real and Complex Analysis. EDIT: Lvriemsurf asked in a comment if we can replace "almost every angle" by "every angle" in the construction. The answer is "no", as follows from the LusinPriwaloff theorems. 

