Khinchin

Notations

Traditional name

Traditional notation

Mathematica StandardForm notation

Primary definition

Specific values

General characteristics

The Khintchine's number K is a constant. It is a positive real number.

Series representations

Generalized power series

Integral representations

On the real axis

Of the direct function

Limit representations

Complex characteristics

Real part

Imaginary part

Absolute value

Argument

Conjugate value

Signum value

Differentiation

Low-order differentiation

Fractional integro-differentiation

Integration

Indefinite integration

Integral transforms

Fourier exp transforms

Inverse Fourier exp transforms

Fourier cos transforms

Fourier sin transforms

Laplace transforms

Inverse Laplace transforms

Inequalities

Theorems

The continued fraction mean theorem

For almost all x∈ℝ:

(lim_(n->∞)) ((((∏_(k=0)^n)(Subscript[q, k])))^(1/n))==K/;        x==(Subscript[q, 0])+(1/((Subscript[q, 1])+(1/((Subscript[q, 2])+(1/((Subscript[q, 3])+…))))))∧ (Subscript[q, k])∈(SuperPlus[ℕ]).

This relation fails for  x=e, rational numbers, solutions of quadratic equations with rational coefficients and quadratic irrationals, such as ϕ, (Sqrt[2]), (Sqrt[3]).

Numerical verifications show that this relation can be valid for x==π, x==ℽ, and x==K.  But it was not accurately proved.

So, K is the limit of the geometric mean of the first n partial quotients of the simple continued fraction for almost any real number x, when n tends to infinity. This limit exists and is a Khinchin constant independent from x for almost all numbers.

History

– A. Khinchin (1934)

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