Somos' quadratic recurrence constant

In mathematical analysis and number theory, Somos' quadratic recurrence constant or simply Somos' constant is a constant defined as an expression of infinitely many nested square roots.

It arises when studying the asymptotic behaviour of a certain sequence[1] and also in connection to the binary representations of real numbers between zero and one.

[2] The constant named after Michael Somos.

It is defined by: which gives a numerical value of approximately:[3] Somos' constant can be alternatively defined via the following infinite product: This can be easily rewritten into the far more quickly converging product representation which can then be compactly represented in infinite product form by: Another product representation is given by:[4] Expressions for

ln ⁡ σ

(sequence A114124 in the OEIS) include:[4][5] Integrals for

ln ⁡ σ

arises when studying the asymptotic behaviour of the sequence[1] with first few terms 1, 1, 2, 12, 576, 1658880, ... (sequence A052129 in the OEIS).

This sequence can be shown to have asymptotic behaviour as follows:[4] Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent

:[6] If one defines the Euler-constant function (which gives Euler's constant for

) as: one has:[7][8][9] One may define a "continued binary expansion" for all real numbers in the set

, similarly to the decimal expansion or simple continued fraction expansion.

This is done by considering the unique base-2 representation for a number

which does not contain an infinite tail of 0's (for example write one half as

Then define a sequence

which gives the difference in positions of the 1's in this base-2 representation.

For example the fractional part of Pi we have:

(sequence A004601 in the OEIS) The first 1 occurs on position 3 after the radix point.

The next 1 appears three places after the first one, the third 1 appears five places after the second one, etc.

By continuing in this manner, we obtain:

(sequence A320298 in the OEIS) This gives a bijective map

, such that for every real number

we uniquely can give:[10]

the limit of the geometric mean of the terms

converges to Somos' constant.

Somos' constant is universal for the "continued binary expansion" of numbers

in the same sense that Khinchin's constant is universal for the simple continued fraction expansions of numbers

The generalized Somos' constants may be given by: for

The following series holds: We also have a connection to the Euler-constant function:[8] and the following limit, where