Problem 523

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claims/: The 1 claim page of Problem 523, one per claimant's result; the problem's standing derives from them.

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Statement. Let $f(z)=\sum_{0\leq k\leq n} \epsilon_k z^k$ be a random polynomial, where $\epsilon_k\in \{-1,1\}$ independently uniformly at random for $0\leq k\leq n$.

Does there exist some constant $C>0$ such that, almost surely,

$$\max_{\lvert z\rvert=1}\left\lvert \sum_{k\leq n}\epsilon_k(t)z^k\right\rvert=(C+o(1))\sqrt{n\log n}?$$

Status. Proved. The answer is yes with $C=1$: Halász proved in 1973 that the unit-circle maximum divided by $\sqrt{n\log n}$ tends almost surely to $1$ (refereed; the accepted claim page is Halász 1973). No formal proof is built or audited in this repository.

Source. erdosproblems.com/523, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #523, https://www.erdosproblems.com/523.

References.

• [Ha73] Halász, G., On a result of Salem and Zygmund concerning random polynomials. Studia Sci. Math. Hungar. (1973), 369-377.
• [SaZy54] Salem, R. and Zygmund, A., Some properties of trigonometric series whose terms have random signs. Acta Math. (1954), 245-301.

Formalization. No formal-conjectures statement; the claim page for Halász 1973 links a Lean 4 proof in the lean-proofs repository at its pinned commit and records what the corpus has and has not checked.

Current assessment

The question (site formulation of 2026-09-04). Whether the maximum modulus on the unit circle of a random polynomial of degree $n$ with independent uniform $\pm1$ coefficients is almost surely $(C+o(1))\sqrt{n\log n}$ for some constant $C>0$. PROVED (the site's label; page last edited 01 February 2026). The site's commentary records that Salem and Zygmund [SaZy54] proved $\sqrt{n\log n}$ to be the right order of magnitude without an asymptotic, and that Halász [Ha73] settled the question with $C=1$.

Standing. One accepted full claim, Halász 1973, refereed in Studia Scientiarum Mathematicarum Hungarica and credited by the site's curator. The account below attributes the $C=1$ answer to Halász's bounds and stated power-polynomial extension. Its coverage is the theorem and extension statements, checked against the paper; the proof is not compiled in this wiki, and no independent proof review is recorded.

Formalization. The site's page shows no formalized statement and no formal-conjectures statement file exists for the problem. A Lean 4 file in the lean-proofs repository declares itself a formalization of Halász's solution and proves the almost-sure limit; the claim page links it at its pinned commit. The development is not built or audited in this repository, so no formalized evidence is listed.

Search scope and read depth. Search scope (2026-10-07): the site's problem page and its empty thread, the formal-conjectures tree and the lean-proofs file. Read depth: Halász 1973 for the theorem, the power-polynomial extension and the received date.

Progress

Halász's theorem is stated first for the random cosine polynomial $f_n(\theta)=\sum_{k=1}^n\epsilon_k\cos(k\theta)$. With probability one, for all sufficiently large $n$,

$$\sqrt{n\log n}-4\sqrt{\frac n{\log n}}\log\log n \leq \max_{0\leq\theta\leq2\pi}|f_n(\theta)| \leq \sqrt{n\log n}+3\sqrt{\frac n{\log n}}\log\log n.$$

The introduction says that the theorem also holds for power polynomials. The final paragraph obtains the same lower bound from the real part of the power polynomial and says that the upper bound follows by using finitely many rotated real parts. Thus the normalized unit-circle maximum tends almost surely to $1$, so the constant in this problem is $C=1$.

Halász indexes the power polynomial by $k=1,\ldots,n$. Relabeling the iid signs and using $n+1$ terms gives the problem's $k=0,\ldots,n$ convention, while $\sqrt{(n+1)\log(n+1)}/\sqrt{n\log n}\to1$.

Known Results

• Halász (1973), printed pp. 369 and 377. The displayed almost-sure bounds above, together with the stated power-polynomial extension, solve the problem with $C=1$. The theorem and extension statements are checked against the paper; the finite-rotation step for the upper bound is the paper's statement, not compiled in this wiki.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• halasz_1973_result_salem_zygmund_concerning_random_polynomials
• halasz_1973_result_salem_zygmund_concerning_random_polynomials / remark_p377
• halasz_1973_result_salem_zygmund_concerning_random_polynomials / theorem_p369
• hayman_lingham_2018_research_problems_function_theory
• hayman_lingham_2018_research_problems_function_theory / problem_4_17