Turan's pure power sum constant

Fact	Value	Source
Best lower bound with an explicit value (constant 42a)	0.5 Biró [Bir94]	Tao's optimization constants
Best upper bound with an explicit value (constant 42a)	0.6873358063* Koshta [Kos2026] — Exact 6873358063/10000000000.	Tao's optimization constants
Best upper bound with an explicit value, independently verified (constant 42a)	0.69368 Harcos [Bir00]	Tao's optimization constants

Description of constant

The constant $C_{42}$ is $lim sup_{n \to \infty} R_{n}$ , where $R_{n} = min 1 \leq k \leq n max 1 \leq i \leq n \sum z_{i}^{k},$ where the minimum is taken over all $z_{1}, \dots, z_{n} \in C$ with $max_{i} ∣ z_{i} ∣ = 1$ .

Known upper bounds

Bound	Reference	Comments
1	Trivial
5/6	Biró [Bir00]
0.69368	Harcos [Bir00]
0.6906538*	Griego [Gri26]	Proposed asymptotic two-block certificate with exact rational interval verification of the limiting inequality. No explicit finite threshold $N$ is supplied.
0.6888863712*	Pinho [Pin26]	Exact $215276991/312500000$ ; 256 uniform middle blocks with $t_{j} = (256 + j) /768$ and $α = 0.64427390171288 + 0.58993506009068 i$ . Rational profile: SHA-256 `705a729155eb508878d17ebea0322627a0ae02451c044a346a99d2e81c7afcab`. Separate interval (100 decimal places, 240 terms) and Arb (768 bits, 400 terms) checks give $∣ Y ∣ / D < 0.688886370983466$ . Unrefereed analytic asymptotic reduction; no explicit finite threshold, optimality or formal verification claimed.
0.6873358063*	Koshta [Kos2026]	Exact $6873358063/10000000000$ . The coefficient-cancellation construction of [Gri26] (with the step profiles of Röhrig, Plata and PR #251), but with the first-block cut-off lowered from $τ = 1/3$ to $τ = 41/200$ . Then up to four corrections reach $b_{n}$ , and the limiting quantity gains the terms $- \frac{1}{6} \sum_{i, j, k} δ_{i} δ_{j} δ_{k} T_{3} + \frac{1}{24} \sum_{i, j, k, l} δ_{i} δ_{j} δ_{k} δ_{l} T_{4}$ , where $T_{d} (u) = \int_{x_{i} \geq u_{i}, \sum x_{i} \leq 1} (1 - \sum x_{i})^{α - 1} / \prod x_{i} d x$ ; the criterion of PR #251 holds verbatim for every $τ \in (0, 1/2)$ with the corresponding $d$ -fold terms ( $d < 1/ τ$ ). 64 uniform middle blocks on $[41/200, 159/200]$ , $α = 0.6928404180017845 + 0.6148849499318954 i$ . Rational profile SHA-256 `be0c406fb2bf58f18a3ecb070daab97b662ec63318d9a83606c2117b8e16cbef`. $T_{4}$ and most $T_{3}$ are evaluated by exact-rational Dirichlet series with explicit tail bounds, the remaining $T_{3}$ by a validated 1-D integral; Arb (384 and 640 bits) gives $∣ Y ∣ / D < 0.68733580619$ , and an independently written checker agrees. Unrefereed analytic asymptotic reduction; no explicit finite threshold, optimality or formal verification claimed.

Known lower bounds

Bound	Reference	Comments
1/6	Atkinson [Atk61]
$1/3$	Atkinson	Mentioned in [Atk69] in a (presumably unpublished) technical report.
$π /8$	Atkinson [Atk69]
1/2	Biró [Bir94]
>1/2	Biró [Bir00b]	Biró's proof delivers some computable constant $1/2 < c < 1/ 2$ , but it is not computed there exactly which.

Additional comments

Computational investigations by Cheer and Goldston [CG96] suggest that $C_{42}$ is close to $0.7$ .
$C_{42}$ is the optimal constant for Erdős problem #519.

References

[Atk61] Atkinson, F. V. On sums of powers of complex numbers. Acta Math. Acad. Sci. Hungar. 12 (1961), 185-188.
[Atk69] Atkinson, F. V. Some further estimates concerning sums of powers of complex numbers. Acta Math. Acad. Sci. Hungar. 20 (1969), 193-210.
[Bir94] Biró, A. On a problem of Tur\'{a}n concerning sums of powers of complex numbers. Acta Math. Hungar. 65 (2000), no. 3, 209-216.
[Bir00] Biró, A. An upper estimate in Tur\'{a}n's pure power sum problem. Indag. Math. (N.S.) 11 (2000), no. 4, 499-508.
[Bir00b] Biró, A. An improved estimate in a power sum problem of Tur\'{a}n. Indag. Math. (N.S.) 11 (2000), no. 3, 343-358.
[CG96] A. Y. Cheer and D. A. Goldston Tur\'{a}n's pure power sum problem. Math. Comp. 65 (1996), no. 215, 1349-1358.
[Gri26] Griego, S. An improved asymptotic certificate for Turan's pure power sum constant $C_{42}$ . GitHub repository, version v1.0.0, commit c8ddce14d9a5e898406d5dc6b8d08bb8a39507c7 (2026). https://github.com/sebastian-griego/turan-c42-certificate/tree/v1.0.0
[Kos2026] Koshta, Dennis, with Claude (Anthropic, Claude Opus 5.5). Lowering the first-block cut-off in the coefficient-cancellation bound for Turán's pure power-sum constant. Unrefereed note, exact certificate and checkers: https://github.com/DennisKoshta/c42-turan-power-sum-certificate (commit 8e1797e0d35454ad18cfebbbeb5858bf202b0748, 23 Oct 2026).

[Pin26] Pinho, R. A 256-step certificate for Turán's pure power-sum constant. Unrefereed note (22 October 2026), paper, exact profile and two separately implemented checkers: https://github.com/rpinho/c42-certificate/tree/a8a9bc309bd1992413fa07595bd60c5e3fa1b65b

What counts as progress

A better upper or lower bound, with a proof or a construction whose value is re-computed by published code (reproducible), ideally with a certificate a deterministic checker can validate.
A formal proof (Lean) of a known bound, or a precise error in a claimed one.
New references for the tables above (literature claims).

Source and licence

Imported from the crowdsourced repository of optimization constants (Terence Tao and contributors), commit 87b4ebbd426f, Apache License 2.0; reformatted for this page. New records should also be reported there.

Current state (tracked automatically)

Nobody has worked on this problem here yet

Current state

The problem