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Statement

Problem 4.17 (p. 76, quoted). "It is shown by Salem and Zygmund [697] that there exist positive constants $C_3,C_4$ such that for every positive $\varepsilon$, we have

$$(C_3-\varepsilon)\big(n\log n\big)^{\frac12}<\max_{|z|=1}|P(z)|<(C_4+\varepsilon)\big(n\log n\big)^{\frac12}$$

apart from $o(2^n)$ polynomials $P(z)$. Is this result true with $C_3=C_4$ and if so, what is the common value?"

The polynomials are those of Problem 4.13 (p. 75), $P(z)=\sum_{k=1}^n\varepsilon_kz^k$ with each $\varepsilon_k=\mp1$, of which there are $2^n$. The book's [697] is R. Salem and A. Zygmund, Some properties of trigonometric series whose terms have random signs, Acta Math. 91 (1954), 245--301. The problem carries no attribution line, and Table 2 (p. 253) lists it among the problems of the 1967 edition.

Update 4.17 (p. 76). The update says Halász (the book's [362]: G. Halász, On a result of Salem and Zygmund concerning random polynomials, Studia Sci. Math. Hungar. 8 (1973), 369--377) proved the conjecture with $C_3=C_4=1$, together with the analogous result for trigonometric polynomials.

Source. W. K. Hayman and E. F. Lingham, Research Problems in Function Theory, arXiv:1809.07200v2 (21 September 2018), Chapter 4, p. 76. The edition read is identified on the source card.

Read depth. Claims checked: the problem, its update and the two cited reference entries were read clause by clause on the printed page. The book proves nothing; it poses and reports.

Proof pointer

None; a problem. Halász's paper is on the Halász card.

Dependencies

None.

Bears on

• Problem 523: Problem 4.17 asks, in the form "apart from $o(2^n)$ polynomials", for a common constant in the $\sqrt{n\log n}$ size of the maximum modulus on the circle; #523 asks for it almost surely, with the sum from $k=0$. Update 4.17 credits Halász with the common value $1$ in the book's form; the problem page records what his paper proves for the almost-sure form.