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Claim. The answer to Problem 519 is yes, with some absolute constant $c>1/2$. A. Biró, An improved estimate in a power sum problem of Turán, Indag. Math. (N.S.) 11 (2000), no. 3, 343--358, doi:10.1016/S0019-3577(00)80003-8, digested on its library card, proves in its Theorem (statement page) that there is an effectively computable absolute constant $q>1/2$ such that for complex $z_1,\ldots,z_n$ with $z_1=1$,

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

The paper computes no value of $q$ and says that the steps of the proof would allow one (p. 344); it argues by contradiction from the assumption that each of the first $n$ power sums has modulus at most $q$, for a $q$ slightly above $1/2$, examining when near-equality could hold in the argument of Biró 1994. The result is a further full proof of the question's statement, strengthening the earlier constants. The proof is not reconstructed in this repository.

Acceptance. Refereed: the paper appeared in Indagationes Mathematicae, a refereed journal, communicated at the meeting of 27 March 2000; the issue head is dated 25 September 2000, which is this page's date. Reviewed: the site's curator, Thomas Bloom, labels the problem PROVED (LEAN) and names this paper as the improvement to an absolute constant above $1/2$ on erdosproblems.com/519 (page last edited 1 February 2026). No independent review is recorded in this repository and none is claimed.

Depends on. Nothing in this wiki: the argument is the paper's own.