Research Problems in Function Theory — Problem 7.3

Definitions and mathematical statement

For a positive integer $n$ and complex numbers $z_1,\ldots,z_n$ with $|z_i|\leq1$, define the power sums \[ S_k=\sum_{i=1}^{n}z_i^k\qquad(k\geq1). \] The tuple may contain repetitions or zeros; the source imposes no distinctness condition.

Question (P. Erdős). Can one have \[ \max_{2\leq k\leq n+1}|S_k|<A^{-n} \] for an absolute constant $A>1$? Absolute means independent of $n$ and the chosen tuple, but the source does not say whether the intended construction is for every sufficiently large $n$, arbitrarily large $n$, or another range. The stated index interval starts at $2$, not $1$.

As written, there is no normalization preventing all $z_i$ from being zero or arbitrarily small, so the intended nontrivial scale is missing. The source separately discusses the normalized condition \[ z_1=1,\qquad |z_i|\leq1\quad(2\leq i\leq n) \] and states that the formula labelled 7.1 can then be satisfied, citing Turán. The text of that labelled formula is not supplied; it cannot safely be identified with the displayed inequality or used to infer a complete answer. A precise nontrivial target therefore requires recovering both the normalization and the quantifiers on $A$ and $n$. No such correction is silently made. The supplied summary reports the question open as of the 2018 edition, with no definitive resolution located.

Short English statement

Ask whether $n$ points in the unit disk can have exponentially small power sums for every exponent from $2$ through $n+1$. The missing scale normalization and the incomplete Turán reference prevent identifying the intended nontrivial version.

Sources

• Hayman, Research Problems in Function Theory (2018), Problem 7.3, attributed to P. Erdős, https://arxiv.org/abs/1809.07200; source-TeX extraction, accessed 2026-07-29. The 2018 update was retained for status triage.
• Turán, cited for the normalized-tuple statement involving source label 7.1. No publication, date, theorem location, or complete referenced formula is supplied.
• Local registry data/unsolvedmath/problems.json, permanent ID $2307003$, source code catalogue entry, Research Problems in Function Theory --- Problem 7.3. The record labels this open and OPEN-TRIAGE, with NEEDS_REVIEW source-status and rights qualifications.