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Irrationality measure of π\pi (C7aC_{7a})

Irrationality measure of π\pi: determine C7aC_{7a} or improve its bounds; the repository records 2 (lower) and 7.103205334137 (upper).

MSC 11J82 · Measures of irrationality and of transcendence

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 ↓
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Problem statement

We define C7aC_{7a} to be the irrationality measure of π\pi:

C7a:=sup⁡μ∈RμC_{7a} := \sup_{\mu\in\mathbb{R}} \mu such that ∣π−p/q∣<q−μ\lvert \pi - p/q \rvert < q^{-\mu} for infinitely many rationals p/qp/q.

Equivalently, C7aC_{7a} is the infimum of all ν\nu such that for every ε>0\varepsilon>0 there exists q0(ε)q_{0}(\varepsilon) with

∣π−pq∣>1qν+ε\left|\pi-\frac{p}{q}\right| > \frac{1}{q^{\nu+\varepsilon}}

for all integers pp and all integers q≥q0(ε)q \ge q_{0}(\varepsilon).

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Background

  • It is widely conjectured that C7a=2C_{7a}=2 (as holds for Lebesgue-a.e. real number), but no improvement over the universal lower bound 22 is currently known.
  • Many upper bounds come from constructing explicit rational approximations to π\pi (often via special integrals or hypergeometric constructions) and then converting these to irrationality-measure estimates; see [B2000] for an accessible overview up to Hata's work.
  • Wikipedia page on irrationality measure
  • Wikipedia page on Dirichlet's approximation theorem

References (from the repository page)

  • [B2000] Beukers, F. A rational approach to π\pi. Nieuw Arch. Wiskd. (5) 1 (2000), no. 4, 372–379.
  • [C1982] Chudnovsky, G. V. Hermite–Padé approximations to exponential functions and elementary estimates of the measure of irrationality of π\pi. In: Lecture Notes in Mathematics 925, Springer (1982), 299–322.
  • [D1842] Dirichlet, L. G. P. Verallgemeinerung eines Satzes aus der Lehre von den Kettenbrüchen nebst einigen Anwendungen auf die Theorie der Zahlen. Sitzungsberichte der Preussischen Akademie der Wissenschaften (1842), 93–95.
  • [H1993] Hata, M. Rational approximations to π\pi and some other numbers. Acta Arith. 63 (1993), no. 4, 335–349.
  • [M1953] Mahler, K. On the approximation of π\pi. Nederl. Akad. Wetensch. Proc. Ser. A 56 = Indag. Math. 15 (1953), 30–42.
  • [Mi1974] Mignotte, M. Approximations rationnelles de π\pi et quelques autres nombres. Mém. Soc. Math. France 37 (1974), 121–132.
  • [RV1993] Rhin, G.; Viola, C. On the irrationality measure of ζ(2)\zeta(2). Ann. Inst. Fourier (Grenoble) 43 (1993), no. 1.
  • [S2008] Salikhov, V. Kh. On the irrationality measure of π\pi. Russian Math. Surveys 63 (2008), no. 3, 570–572.
  • [ZZ2020] Zeilberger, D.; Zudilin, W. The irrationality measure of π\pi is at most 7.103205334137…7.103205334137\ldots. Moscow Journal of Combinatorics and Number Theory 9 (2020), no. 4, 407–419. arXiv:1912.06345

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 2; best known upper bound 7.103205334137. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
4242 [M1953] First proof that C7aC_{7a} is finite (i.e. π\pi is not a Liouville number).
20.620.6 [Mi1974] Improves Mahler's exponent.
19.889994419.8899944 [C1982] Uses Hermite–Padé approximation methods.
14.79707414.797074 [RV1993] Follows from an effective irrationality measure for ζ(2)=π2/6\zeta(2)=\pi^2/6.
8.0160458.016045 [H1993] Record for many years (Hata gave a series of improvements).
7.6063087.606308 [S2008] Salikhov's bound.
7.1032053341377.103205334137 [ZZ2020] Current record bound.

Known lower bounds

Bound Reference Comments
22 [D1842] Dirichlet's theorem; holds for every irrational number (in particular for π\pi).
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Irrationality measure of $\pi$ ($C_{7a}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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