Somos' quadratic recurrence constant

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Somos’ quadratic recurrence constant, usually denoted by σ, is a number that shows up in several areas of analysis and number theory. It can be defined in several equivalent ways, and all give the same value, which is approximately 1.6616879496.

A simple way to think about it
- Continued binary expansion viewpoint: Take any x in (0,1]. Write x in a special binary form that records the gaps between consecutive 1s in its binary expansion. If you let a1, a2, a3, … be those gap lengths, then for almost every x the geometric mean of the first n gaps tends to a constant σ as n grows:
σ = lim as n → ∞ of (a1 a2 … an)^(1/n).
- This behavior is universal in the same spirit as Khinchin’s constant is universal for simple continued fractions.

Other ways to see σ
- Nested square roots and infinite products: σ can be expressed by infinite nested square-root expressions and by rapidly converging infinite products. These provide different but equivalent ways to define σ.
- Connections to special functions: There are representations of ln σ in terms of derivatives of the Lerch transcendent and related functions, linking σ to deeper objects in analysis.

A link to a growing sequence
- σ also appears when studying the asymptotics of a fast-growing sequence tied to a quadratic recurrence (the Somos-4 type sequence) that begins 1, 1, 2, 12, 576, 1658880, … Its long-term growth is governed by σ in a precise way.

Generalizations and related ideas
- Generalized Somos constants: For parameters t > 1, there are generalized constants defined in analogous ways, with their own series and product representations.
- Links to Euler’s constant: There are integral and limit formulas that connect σ to the Euler–Mascheroni constant and related constant-function identities, showing unexpected connections in number theory.

In short
σ is a universal constant associated with the continued-binary expansion of numbers in (0,1], and it also appears in the study of certain quadratic recurrences and related representations. It has multiple equivalent definitions, including nested radical and product forms, and rich connections to special functions and fundamental constants.