Problem 519

Let $z_1,\ldots,z_n\in \mathbb{C}$ with $z_1=1$ . Must there exist an absolute constant $c>0$ such that

\max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c?

Source: erdosproblems.com/519

Analysis

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN). The site labels the problem PROVED (LEAN) (page last edited 1 February 2026), credits Atkinson [At61b] with the solution, $c=1/6$ , and names Biró's improvements to $c=1/2$ [Bi94] and to an absolute constant above $1/2$ [Bi00]; the Lean qualifier refers to formalizations of Atkinson's proof that have not been built here (see Formalization). Three accepted claim pages record three full proofs, each on its refereed venue and the site's credit: Atkinson 1961, Biró 1994 and Biró 2000; the frontmatter standing derives from them.

At61b: Atkinson, F. V., On sums of powers of complex numbers. Acta Math. Acad. Sci. Hungar. 12 (1961), 185--188.
Bi00: Biró, András, An improved estimate in a power sum problem of Turán. Indag. Math. (N.S.) 11 (2000), no. 3, 343--358.
Bi00b: Biró, A., An upper estimate in Turán's pure power sum problem. Indag. Math. (N.S.) 11 (2000), no. 4, 499--508.
Bi94: Biró, A., On a problem of Turán concerning sums of powers of complex numbers. Acta Math. Hungar. 65 (1994), no. 3, 209--216.

Problem 519

Status

Proof claims

Discussion