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Khinchin

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

Khinchin

is Khinchin's constant K (also Khintchine's constant), with numerical value ~= 2.68545.

NumericQ[Khinchin]

is True, and D[Khinchin, x] is 0. N[Khinchin, prec]

Notes Khinchin's constant is the limiting geometric mean of the partial quotients in the continued-fraction expansion of almost every real number, given by the product over s \>= 1 of (1 + 1/(s (s + 2)))^Log2\[s\]. It is a mathematical constant: it has attributes Constant and Protected, evaluates it to any precision.

Examples (4)

Every input below was run against the current Mathilda build and its output recorded.

Applications (4)

In[1]:= N[Khinchin]
Out[1]= 2.68545

In[2]:= N[Khinchin, 60]
Out[2]= 2.685452001065306445309714835481795693820382293994462953051151

In[3]:= NumericQ[Khinchin]
Out[3]= True

In[4]:= D[Khinchin, x]
Out[4]= 0

Options & behaviour

The constant values for GoldenAngle, Glaisher, and Khinchin live in the numeric constant table (src/numeric.c); their MPFR fillers compute GoldenAngle from its closed form, and Glaisher/Khinchin from the series above. Their Constant/Protected attributes are stamped in numeric_init.

Implementation notes

Definition. Khinchin is Khinchin's constant K (also "Khintchine's constant") ≈ 2.6854520011: the geometric mean of the partial quotients in the continued-fraction expansion of almost every real number, Product_{s>=1} (1 + 1/(s (s + 2)))^Log2[s]. It is a constant symbol: row { "Khinchin", 2.6854520011…, fill_mpfr_khinchin } of kConstants[] in src/numeric.c supplies its value and the numericalize_init loop stamps it Constant | Protected, so Attributes[Khinchin] is {Constant, Protected}. Interned name SYM_Khinchin (src/sym_names.c), docstring in src/info.c. No DownValues; D[Khinchin, x] -> 0 follows from Constant.

Representation & numeric value. A bare EXPR_SYMBOL, kept exact under evaluation. N[Khinchin] returns the tabled machine double. N[Khinchin, k] runs fill_mpfr_khinchin, which uses the geometrically convergent Bailey–Borwein–Crandall zeta series Log[K] Log[2] = Sum_{n>=1} ((Zeta[2n] - 1)/n) Sum_{k=1}^{2n-1} (-1)^(k+1)/k: Zeta[2n]-1 ~ 4^-n, so ~bits/2 terms (each via mpfr_zeta_ui, with a running alternating-harmonic inner sum) reach the target; K = Exp[S / Log[2]] at bits + 64 guard precision. Machine value 2.68545; N[Khinchin, 50] = 2.68545200106530644530971483548179569382038229399446.

Usage & limits. Khinchin is the metric constant of continued fractions; NumericQ[Khinchin] is True (whitelisted in is_numeric_quantity, src/core.c). Whether K is irrational is open. The implementation limit mirrors Glaisher's: the filler is a bespoke convergent series rather than a library constant, with a term cutoff at 2^-(bits+32).

  • Attributes Constant, Protected. Attributes[Khinchin] = {Constant, Protected}; the symbol cannot be reassigned.
  • Propagated as an exact, unevaluated symbol; NumericQ[Khinchin] is True and D[Khinchin, x] = 0.
  • N[Khinchin] gives the machine value 2.68545; N[Khinchin, prec] gives any precision, e.g. N[Khinchin, 50] = 2.6854520010653064453097148354817956938203822939945.

Arbitrary precision uses the geometrically convergent zeta series ln K · ln 2 = Σ_{n>=1} (ζ(2n) − 1)/n · Σ_{k=1}^{2n−1} (−1)^(k+1)/k (the Bailey–Borwein–Crandall form). Verified to 250 digits.

Attributes: Constant, Protected.

References

See also: GoldenAngle, Glaisher

Notes & additional examples

Notes

Khinchin is Khinchin's (Khintchine's) constant K ~= 2.68545, the limiting geometric mean of the partial quotients in the continued-fraction expansion of almost every real number: K = Product[(1 + 1/(s (s + 2)))^Log2[s], {s, 1, Infinity}]. It carries the Constant and Protected attributes, so it stays symbolic until N[Khinchin, prec] evaluates it to the requested precision.