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Claim. The answer to Problem 519 is yes, with $c=1/2$. András Biró, On a problem of Turán concerning sums of powers of complex numbers, Acta Math. Hungar. 65 (1994), no. 3, 209--216, doi:10.1007/BF01875148, digested on its library card, proves in Theorem 1 that for arbitrary complex $z_1,\ldots,z_n$ with $z_1=1$,

$$\max_{1\le j\le n}\Bigl\lvert\sum_{t=1}^n z_t^j\Bigr\rvert>\frac12.$$

This is an independent proof of the question's statement, with a constant three times Atkinson's bound of 1961; the paper's introduction records Atkinson's own later improvements, to $1/3$ and then to $\pi/8$ for $n<1600$ and to a constant $s_0<\pi/8$ for all large $n$, and presents Theorem 1 as an improvement on them. The proof forms the polynomial with roots $z_2,\ldots,z_n$, applies two Newton--Girard identities to its coefficients, and uses a planar dichotomy (Lemma 1) under which each new coefficient either makes a Newton--Girard expression large or forces the consecutive coefficient partial sums to grow. The complete published proof is reconstructed in this repository at Theorem 1 with its planar input at Lemma 1; the reconstruction is compilation, not independent review, and the value $1/2$ is not claimed to be sharp. Biró's later paper Biró 2000 improves the constant.

Acceptance. Refereed: the paper appeared in Acta Mathematica Hungarica, a refereed journal; the publisher's record (Crossref) dates the issue September 1994, and this page's date is the first day of that month. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED (LEAN) and names this paper as the improvement of the constant to $1/2$ on erdosproblems.com/519 (page last edited 1 February 2026). The proof has been reconstructed here but not independently reviewed, and no review verdict is claimed.

Depends on. Nothing in this wiki: the argument is the paper's own; the reconstruction lives in the library.