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Halász [Ha73] answers the question of Salem and Zygmund, which Hayman's collection poses for power polynomials as Problem 4.17, by proving that the normalized maximum of a random sign polynomial has an almost-sure limit, and that the limit is Salem and Zygmund's upper constant. His theorem is stated for the cosine polynomial $f_n(\theta)=\sum_{k=1}^n\epsilon_k\cos(k\theta)$ with independent uniform signs: with probability one, for all sufficiently large $n$,

$$\sqrt{n\log n}-4\sqrt{\frac n{\log n}}\log\log n \le\max_{0\le\theta\le2\pi}|f_n(\theta)| \le\sqrt{n\log n}+3\sqrt{\frac n{\log n}}\log\log n.$$

The introduction says that the theorem also holds for power polynomials, and the closing paragraph gives the transfer: the lower bound holds for $P_n(z)=\sum_{k=1}^n\epsilon_kz^k$ on $|z|=1$ because $f_n$ is the real part of $P_n(e^{i\theta})$, and the upper bound follows by applying the argument to the real parts of finitely many fixed rotations of $P_n$. Hence the unit-circle maximum divided by $\sqrt{n\log n}$ tends almost surely to $1$, which answers Problem 523 yes with $C=1$; the shift from Halász's $n$ terms indexed from $1$ to the problem's $n+1$ terms indexed from $0$ changes nothing, since $\sqrt{(n+1)\log(n+1)}/\sqrt{n\log n}\to1$. The proof treats the lower bound first, through a smooth cutoff written as a Fourier–Stieltjes transform, sums over equally spaced points and Chebyshev's inequality; the source card digests the paper. The page's date is the day the paper was received by the journal, 1973-04-20, as its last page records.

Depends on. Nothing in this wiki; the result rests on the refereed paper linked above.

Acceptance. Refereed: the paper appeared in Studia Scientiarum Mathematicarum Hungarica 8 (1973), 369–377, in the volume whose repository scan the paper link names. Reviewed: the site's curator, Thomas F. Bloom, records the problem as settled by Halász with $C=1$ in the problem's commentary (page last edited 01 February 2026). The theorem and extension statements are checked against the paper; the finite-rotation step for the upper bound of the power polynomial is the paper's statement, not compiled in this wiki. Formalization: a Lean 4 file in the lean-proofs repository, linked above at its pinned commit, declares itself a formalization of Halász's solution, names Codex and GPT-5.6 Sol as its formal authors, and proves erdos_523: on the product space of independent uniform signs, almost surely the maximum modulus divided by $\sqrt{n\log n}$ tends to $1$; the file prints the axioms of erdos_523 after the theorem. This corpus has neither built that development nor audited its statement, so no formalized evidence is listed.