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Source. Theorem, printed p. 344 (physical PDF p. 2); proof in sections 2--4, pp. 345--357. Read on the page image; the text layer is machine OCR.

Statement

One effectively computable absolute constant $q>\tfrac12$ serves every $n$: whenever $z_1,\dots,z_n$ are complex numbers with $z_1=1$,

$$\max_{1\le j\le n}|S_j|>q,\qquad S_j=z_1^j+\cdots+z_n^j.$$

Hence $R_n>q$ for every $n$, where $R_n$ is the minimum of $\max_{1\le j\le n}|S_j|$ over $n$-tuples with $\max_t|z_t|=1$ (p. 343; the normalization $z_1=1$ is equivalent).

The paper does not compute a concrete value of $q$; it says this "would be possible following the steps of our proof" and that determining the best constant obtainable by its ideas seems rather complicated (p. 344).

Proof, as a pointer

The proof (pp. 344--357) assumes $|S_j|\le q$ for $1\le j\le n$ with $1/2<q<q_0<1/\sqrt2$ fixed, works with the Newton--Girard relations (3), (4) between the $S_j$ and the coefficients $b_t$ of $\prod_{t=2}^n(Z-z_t)$, adds formulas (14) obtained by summing (3), and shows that the near-equality forced in the 1994 argument is impossible: for $\alpha<\pi/4$ close enough to $\pi/4$ and then $q>1/2$ close enough to $1/2$, inequality (47) on p. 357 is contradictory. The proof was not checked here.

Bears on. #519; the Theorem proves the problem's existence statement with an effectively computable absolute constant $q>1/2$, whose value the paper does not compute.