AI-handled content

This site is generated and maintained by AI and may be prone to errors. Please verify any claim independently before relying on it.

Lévy's Constant and Khinchin's Constant: How the Typical Continued Fraction Computes the Circle Constant as π²/(12 ln 2)¶

Ask a computer for the digits of π and it hands you a string. Ask it for the continued fraction of π and it hands you something stranger:

\[3 + \cfrac{1}{7 + \cfrac{1}{15 + \cfrac{1}{1 + \cfrac{1}{292 + \cdots}}}} = [3;\,7,\,15,\,1,\,292,\,1,\,1,\,1,\,2,\,1,\,3,\,1,\,14,\,2,\ldots]\]

That lone \(292\) is why \(\tfrac{355}{113}\) is such a famously good approximation to the circle constant — the next term after a huge one is always tiny, and a tiny term means the convergent is already as good as it can be. But the continued fraction of a specific number is not what this article is about. It is about the continued fraction of a typical number, the statistics that hold for almost every real, and the two exact constants — Lévy's and Khinchin's — that summarise those statistics. One of them is built, digit for digit, out of the circle constant. The other, remarkably, contains no trace of it. Both are computed, never measured.

The four questions¶

As with every entry in this series, the same four questions are put to the subject: where exactly does the circle constant enter, is it computed or measured, does it cancel anywhere, and what changes when the constant is the golden value \(\hat\pi = 4/\sqrt\varphi\) instead of the analytic \(3.14159\ldots\)? Here the answers are unusually clean — because the whole theory rests on one integral, \(\int_0^1 \frac{\ln x}{1+x}\,dx = -\frac{\pi^2}{12}\), and on one product that has no circle constant in it at all.

Continued fractions in one paragraph¶

Every real number \(x\) has a continued fraction

\[x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cdots}}} = [a_0;\,a_1,\,a_2,\,a_3,\ldots],\]

with integer partial quotients \(a_0, a_1, a_2,\ldots\) (\(a_n \ge 1\) for \(n \ge 1\)). Truncating at the \(n\)-th term gives a convergent \(p_n/q_n\) — the best rational approximation to \(x\) with denominator at most \(q_n\). The engine behind all of this is the Gauss map

\[T(x) = \frac{1}{x} \bmod 1,\]

which shifts the continued-fraction expansion one place to the left: if \(x = [a_0; a_1, a_2,\ldots]\) then \(T(x) = [a_1; a_2, a_3,\ldots]\). The partial quotients are simply the iterates \(a_n = \lfloor T^{n-1}(x)\rfloor\) (after the first). Everything that follows is a statement about the long-run behaviour of this map.

The Gauss map has a measure, and Gauss knew it¶

In a letter to Laplace around 1812, Gauss observed that for almost every real \(x\) (in the sense of Lebesgue measure — every \(x\) except a set of measure zero) the partial quotients are distributed according to

\[\mu(a_n = k) = -\log_2\!\left(1 - \frac{1}{(k+1)^2}\right) = \log_2\frac{(k+1)^2}{k(k+2)}.\]

This is the Gauss–Kuzmin distribution. The first few values are genuinely informative about how lopsided it is:

\(k\)	\(\mu(a=k)\)
1	0.415037499278844
2	0.169925001442312
3	0.093109404391482
4	0.058893689053569
5	0.040641984913019
10	0.011972634221981

More than two fifths of all partial quotients in a typical expansion are the number \(1\); the average is infinite in the naive sense, because \(\sum_k k\,\mu(k)\) diverges — yet the logarithmic average is perfectly finite and exact, which is the whole story. The measure \(\mu\) is the invariant measure of the Gauss map, its density \(g(x) = \frac{1}{(1+x)\ln 2}\) on \([0,1)\), and the theorem that the empirical counts converge to it is the Gauss–Kuzmin–Lévy theorem. Kuzmin (1928) gave the first geometric rate of approach, Lévy (1929) sharpened it, and Wirsing (1974) gave the best-known refinement. As with every quadrature in this series: it is a computed limiting average of an iteration, not a measurement of anything physical.

Lévy's constant: the price of a digit¶

The convergents' denominators do not grow by a constant ratio — they grow by a constant exponential rate. Lévy's theorem states that for almost every \(x\),

\[\lim_{n\to\infty} q_n^{1/n} = e^{\lambda}, \qquad \lambda = \frac{\pi^2}{12\ln 2} = 1.1865691104156254528\ldots\]

so that

\[e^{\lambda} = e^{\pi^2/(12\ln 2)} = 3.275822918721811159787681882453843863608\ldots\]

This is Lévy's constant: the amount each additional continued-fraction term multiplies the denominator by, on average. Ten terms buy you a denominator roughly \(3.2758^{10} \approx 1.5\times10^{5}\); twenty buy about \(2.3\times10^{10}\); a hundred buy a denominator with about fifty decimal digits. The constant is a rate of growth, exact, universal, and the same for almost every number.

Where π enters: it is ζ(2), computed¶

The circle constant enters through exactly one integral. Write \(\lambda\) as the logarithmic average of a partial quotient under the Gauss–Kuzmin measure, and the computation reduces to

\[\int_0^1 \frac{\ln x}{1+x}\,dx = -\eta(2) = -\frac{\pi^2}{12},\]

where \(\eta\) is the Dirichlet eta function, the alternating Basel sum

\[\eta(2) = 1 - \frac{1}{2^2} + \frac{1}{3^2} - \frac{1}{4^2} + \cdots = \frac{\pi^2}{12} = 0.82246703342411321824\ldots\]

So \([\pi]^2/12\) is not smuggled in and it is not fitted: it is the evaluated limit of an alternating series, the same kind of computation that gives \(\zeta(2) = \pi^2/6\) in the Basel problem. Expanded with the Gauss-map density,

\[\lambda = -\frac{1}{\ln 2}\int_0^1 \frac{\ln x}{1+x}\,dx = \frac{\pi^2}{12\ln 2} = 1.186569110415625\ldots, \qquad e^{\lambda} = 3.275822918721811\ldots\]

The constant is in the exponent, sitting on a \(\pi^2\), and this placement controls everything Golden Pi will do to it below.

The π-free twin: Khinchin's constant¶

Now the surprise. The other famous constant of typical continued fractions is Khinchin's constant (Khinchin, 1934): for almost every \(x\),

\[\lim_{n\to\infty}\left(a_1 a_2 \cdots a_n\right)^{1/n} = K_0 = \prod_{n=1}^{\infty}\left(1 + \frac{1}{n(n+2)}\right)^{\log_2 n} = 2.685452001065306445309714835481795693820\ldots\]

Look at that product: it is built entirely out of the integers \(n\) and the number \(2\) (inside the logarithm base). There is no circle constant in it. The product converges with maddening slowness, which is why \(K_0\) is quotable only by its computed value — the partial products are

\(N\)	\(\prod_{n=1}^{N}\left(1+\frac{1}{n(n+2)}\right)^{\log_2 n}\)
100	2.479449507190304
1 000	2.655030716354816
10 000	2.681499686663012
100 000	2.684967264898291
\(\to\infty\)	2.685452001065306…

This is the honest heart of the article. Lévy's constant carries \(\pi\) in its exponent; the Gauss–Kuzmin entropy carries \(\pi\); Lochs' constant carries \(\pi\) — and Khinchin's constant carries none. The circle constant is not welded to the theory of continued fractions; it appears in a specific, replaceable coefficient (\(\eta(2)\)), and one of the theory's two flagship constants does not touch it.

There is a second, sharper honesty point. Khinchin's constant is proven for almost every number — the exceptional set has measure zero — and has never been proven for \(\pi\), for \(e\), for \(\sqrt2\), or for any number a human has ever actually cared about. The same pattern in \(\pi\)'s own expansion is therefore a conjecture, not a theorem.

Lochs' theorem: how many terms buy a decimal digit¶

Gustav Lochs (1964) connected continued fractions to ordinary decimal accuracy. If \(n\) decimal digits of accuracy require \(m\) continued-fraction terms, then for almost every \(x\),

\[\lim \frac{n}{m} = \frac{6\ln 2\,\ln 10}{\pi^2} = 0.9702701143920339257402560192100108337813\ldots\]

or inverted, \(1/0.970270114\ldots = 1.0306408341007129\) — about 1.03 continued-fraction terms buy one decimal digit. Converted to bits, the same statement reads that one term carries \(\log_2(10)\times0.97027 = 3.223\) bits of information about \(x\), matching the Gauss–Kuzmin source entropy \(H\) below.

π's own continued fraction — and the honest caveat¶

Here is the continued fraction of the circle constant itself, to enough terms to be interesting:

\[[3;\,7,\,15,\,1,\,292,\,1,\,1,\,1,\,2,\,1,\,3,\,1,\,14,\,2,\,1,\,1,\,2,\,2,\,2,\,2,\,1,\,84,\,2,\,1,\,1,\,15,\,3,\,13,\,1,\,4,\ldots]\]

The term \(292\) is famously anomalous, and it is precisely why \(355/113\) is so good: from it, \(355/113 = 3.141592920353982\ldots\), correct to six decimal places. But the statistics of a finite prefix wander wildly, and this is worth watching concretely. The geometric mean of the first \(n\) partial quotients of \(\pi\) (skipping the leading \(3\)) is

\(n\)	geometric mean of \(a_1\ldots a_n\)
10	3.361030
20	2.487712
30	2.886591
40	3.147711
59	2.750654

It is nowhere near \(K_0 = 2.685452\) after these many terms — and it need not be. Convergence to the Khinchin constant is glacial, and for \(\pi\) it is not even known to occur. Nothing on this page allows a reader to claim that \(\pi\)'s continued fraction measures the circle constant: the partial quotients are integers, produced by a deterministic algorithm, and the constants they are supposed to approach are limiting averages that no finite computation reaches.

What Golden Pi changes¶

The golden constant is \(\hat\pi = 4/\sqrt\varphi\), the root of \(x^4 + 16x^2 - 256 = 0\), with the exact square

\[\hat\pi^2 = \frac{16}{\varphi} = 8(\sqrt5 - 1) = 9.888543819998317571273389349850209883525\ldots\]

against \(\pi^2 = 9.869604401089358618834490999876151135314\ldots\). Every quantity in this article carries \(\pi\) squared, so the recurring one-power gap of \(0.0959022309\%\) doubles to \(0.1918964\%\), undiluted:

quantity	conventional	Golden Pi	shift
\(\pi^2/12 = \eta(2)\)	0.8224670334241132182	\(\frac{4}{3\varphi} = 0.8240453183331931309\)	\(+0.1918964\%\)
Lévy exponent \(\lambda = \pi^2/(12\ln 2)\)	1.1865691104156254528	\(\frac{4}{3\varphi\ln 2} = 1.1888460942270649314\)	\(+0.1918964\%\)
Lévy's constant \(e^{\lambda}\)	3.2758229187218111598	\(e^{4/(3\varphi\ln 2)} = 3.2832904129322327269\)	\(+0.2279578\%\)
Gauss–Kuzmin entropy \(H = \pi^2/(6\ln 2)\)	2.3731382208312509056	\(\frac{8}{3\varphi\ln 2} = 2.3776921884541298627\)	\(+0.1918964\%\)
Lochs' constant \(6\ln2\ln10/\pi^2\)	0.9702701143920339257	0.9684117667439049437	\(-0.1915289\%\)
Khinchin's constant \(K_0\)	2.6854520010653064453	2.6854520010653064453	none

The Lévy constant shifts by more than the others — \(0.2279578\%\) rather than \(0.1918964\%\) — and for an exact reason: it sits in an exponent. If \(\text{Lévy} = e^{\lambda}\) with \(\lambda \propto \pi^2\), then the relative change is \(\lambda\) times the exponent's relative change, and \(\lambda = 1.1865691\) gives

\[\frac{\Delta \text{Lévy}}{\text{Lévy}} = \lambda \times 0.1918964\% = 1.1865691 \times 0.001918964 = 0.002279578.\]

That is precisely the \(+0.2279578\%\) in the table — the extra size is the amplified leverage of an exponent, not a new effect. Khinchin's constant, having no circle constant inside it, does not move by a single digit under the relabel, which is the cleanest possible statement of where the constant lives and where it does not.

The exact golden relations¶

Where the relabel produces exact algebraic forms rather than shifted decimals, they are credited plainly:

\[\hat\pi^2 = \frac{16}{\varphi}, \qquad \frac{\hat\pi^2}{16} = \frac{1}{\varphi} = 0.6180339887498948482\ldots \text{ exactly}, \qquad \text{Lévy}_{\text{golden}} = 2^{\,4/(3\varphi(\ln 2)^2)} = e^{\,4/(3\varphi\ln 2)}.\]

And the recurring geometric exactness of the golden value holds here too: a disk of radius \(\varphi^{1/4} = 1.1278384855616822603\) has area exactly \(4\) under Golden Pi, where the conventional value gives \(3.996167586135262\ldots\) — short by exactly the gap.

Computed, never measured¶

Nothing on this page is a measurement, and it is worth saying why explicitly. The quantity \(\pi^2/12\) is the evaluated limit of the alternating series \(\eta(2)\); the Gauss–Kuzmin density is the computed invariant measure of an iteration; Lévy's constant is the exponential of a computed integral; Khinchin's constant is the limit of a computed product. Every one of them is a limit of a computation, fixed by arithmetic alone, with no physical input anywhere.

This has a consequence for the π dispute that the site always states plainly. The recurring \(0.0959022309\%\) gap between \(\pi\) and \(\hat\pi\) is not something a laboratory could settle here, and not because the equipment is too coarse — but because there is no measurable quantity in the theory of typical continued fractions at all. One cannot measure the limiting geometric mean of the infinite partial-quotient sequence of a typical real number, because "typical" is a statement about almost every number, not about a physical object. The comparison is decided the only way it can be: by arithmetic, in the open, on an exact series and an exact integral. Contrast with a genuine measurand — the fine-structure constant, or a physical length — where the apparatus does arbitrate. Here it does not, and no honest ledger pretends otherwise.

Computed, never measured.