OpenFrontiers
Join
← Back to the index
OP-12006OpenNumber theory

Classical zero-free region constant (C8C_{8})

Classical zero-free region constant: determine C8C_{8} or improve its bounds; the repository records 0.755106 (lower) and 4.896 (upper).

MSC 11M26 · Nonreal zeros of ζ(s)\zeta (s) and L(s,χ)L(s, \chi); Riemann and other hypotheses

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
Read original source ↗ 0Submit progressLean 4 ↓Discussion (0)

Problem statement

C8=RC_{8} = R is the least constant such that there are no zeroes σ+it\sigma+it of the Riemann zeta function with ∣t∣≥2\lvert t \rvert \geq 2 and σ>1−1Rlog⁡∣t∣\sigma > 1 - \frac{1}{R \log \lvert t \rvert}.

Lean 4

Not yet attached

No Lean statement or proof is attached yet. Have a formalization? Share its source to add it to this problem.

Share a formalization →Verification guide ↗

Background

  • This constant is relevant to the classical error term in the prime number theorem; in particular, π(x)=Li(x)+O(xexp⁡(−log⁡x/R))\pi(x) = \mathrm{Li}(x) + O\left(x \exp\left(-\sqrt{\log x / R}\right)\right).

References (from the repository page)

  • [F2002] Ford, Kevin. Vinogradov's integral and bounds for the Riemann zeta function. Proceedings of the London Mathematical Society, 85(3):565-633, 2002.
  • [F2002b] Ford, Kevin. Zero-free regions for the Riemann zeta function. In Number Theory for the Millennium, II (Urbana, IL, 2000), pages 25-56. A K Peters, 2002.
  • [JK2014] Jang, Won-Jin; Kwon, Seunghyun. A note on Kadiri's explicit zero free region for Riemann zeta function. Journal of the Korean Mathematical Society, 51(6):1291-1304, 2014.
  • [K2005] Kadiri, Habiba. Une région explicite sans zéros pour la fonction ζ de Riemann. Acta Arithmetica, 117(4):303-339, 2005.
  • [K1977] Kondrat'ev, V. P. Some extremal properties of positive trigonometric polynomials. Mathematical notes of the Academy of Sciences of the USSR, 22(3):696-698, 1977.
  • [MT2014] Mossinghoff, Michael J.; Trudgian, Timothy S. Nonnegative trigonometric polynomials and a zero-free region for the Riemann zeta-function. Journal of Number Theory, 157:329-349, 2015.
  • [MTY2022] Mossinghoff, Michael J.; Trudgian, Timothy S.; Yang, Andrew. Explicit zero-free regions for the Riemann zeta-function. arXiv preprint arXiv:2212.06867, 2022.
  • [BTY2026] Bellotti, Chiara; Trudgian, Timothy; Yang, Andrew. Zero-free regions inspired by work of Heath-Brown. arXiv:2603.21490 (2026).
  • [R1941] Rosser, J. Barkley. Explicit bounds for some functions of prime numbers. American Journal of Mathematics, 63(2):211-232, 1941.
  • [RS1962] Rosser, J. Barkley; Schoenfeld, Lowell. Approximate formulas for some functions of prime numbers. Illinois Journal of Mathematics, 6(1):64-94, 1962.
  • [RS1975] Rosser, J. Barkley; Schoenfeld, Lowell. Sharper bounds for the Chebyshev functions θ(x) and ψ(x). Mathematics of Computation, 29(129):243-269, 1975.
  • [S1970] Stechkin, S. B. Zeros of the Riemann zeta-function. Mathematical notes of the Academy of Sciences of the USSR, 8(4):706-711, 1970.
  • [dlVP1899] de la Vallée Poussin, Charles-Jean. Sur la fonction ζ (s) de Riemann et le nombre des nombres premiers inférieurs à une limite donnée. Mémoires de l'Académie royale de Belgique, 59:1-74, 1899.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 0.755106; best known upper bound 4.896. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
34.8234.82 [dlVP1899] Implies the prime number theorem
1919 [RS1962]
9.645919.64591 [S1970]
8.4638.463 [F2002]
5.696935.69693 [K2005]
5.683715.68371 [JK2014]
5.56663055.5666305 [MT2014]
5.5586915.558691 [MTY2022]
4.8964.896 [BTY2026] Bellotti–Trudgian–Yang: ζ(σ+it)≠0\zeta(\sigma+it)\ne 0 for t≥3t\ge 3 and σ>1−1/(4.896log⁡t)\sigma > 1 - 1/(4.896\log t); extends to ∣t∣≥2\lvert t\rvert \ge 2 using verified RH up to H=3⋅1012H = 3\cdot 10^{12}. A companion Theorem 2 sharpens the constant further to 4.85944.8594 conditionally on unpublished improvements to prior inputs.

Known lower bounds

Bound Reference Comments
2/log⁡γ1≈0.7551062/\log \gamma_{1} \approx 0.755106 - Unconditional: γ1=14.1347…\gamma_{1} = 14.1347\dots is the ordinate of the lowest zero of ζ\zeta, and that zero has σ=1/2\sigma = 1/2, so any admissible RR must satisfy 1/2≤1−1/(Rlog⁡γ1)1/2 \leq 1 - 1/(R\log\gamma_{1}). Optimal assuming RH.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Classical zero-free region constant ($C_{8}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

Recorded progress

Add yours →

Nothing recorded yet. A pointer to a paper the entry is missing counts.

Machine attempts

0 recorded

Attempts run against the statement below and are recorded in full, including the failures. The point is to stop the next model — and the next person — from re-walking a road that is already known to end. Each attempt costs 25 credits.

No machine attempts recorded.

Sign in to run an attempt. New accounts start with enough credits for ten.

Discussion

0 notes

Notes are for clarifying the statement, flagging a missing reference, or recording why an approach fails. Claimed solutions belong in a submission, where they can be reviewed.

No notes yet.

Sign in to add a note.