math.NT — Number Theory

Turán's pure power sum problem: certified bounds for finite $n$ and $\limsup R_n\le 0.68637$

Contributed by Vasileios Bakeas, Ricardo Pinho

For $n\ge1$ let $R_n=\inf\max_{1\le k\le n}|z_1^k+\dots+z_n^k|$, the infimum being over all complex $z_1=1,z_2,\dots,z_n$. This is Turán's pure power sum problem (Erdős problem #519). The best published upper bound for the constant $\limsup R_n$ is $0.69368$, from a computation of Harcos reported by Biró in 2000. We prove $\limsup_{n\to\infty}R_n\le0.68637$. The construction is the Biró–Harcos linearization, with the first half of the power sums prescribed as $s_k=Te^{i\varphi(k/n)}$ for a piecewise-linear phase $\varphi$ that varies from the start. We prove a limit theorem: the quantity that controls the second half converges to an explicit functional of the solution of a linear Volterra equation whose associated operator is a contraction. The resulting inequality is certified in ball arithmetic by an a posteriori error bound, with two separately written programs that follow the same mathematical argument. We also give rigorous upper bounds for $R_n$ at 13 values of $n$ between $300$ and $10^6$, from explicit exact certificates, each checked by one or two programs written independently of the generator. In an appendix, R. Pinho applies the limit theorem to a finer profile, with sharper error bounds, and certifies $\limsup R_n\le0.6863607950$. The work was done with substantial help from AI models; it has not been refereed.