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 and also in connection to the binary representations of real numbers between zero and one. The constant named after Michael Somos. It is defined by:
σ = 1 2 3 4 5 ⋯ {\displaystyle \sigma ={\sqrt {1{\sqrt {2{\sqrt {3{\sqrt {4{\sqrt {5\cdots }}}}}}}}}}}
which gives a numerical value of approximately:
σ = 1.661687949633594121295 … {\displaystyle \sigma =1.661687949633594121295\dots \;} (sequence A112302 in the OEIS).
Sums and products Somos' constant can be alternatively defined via the following infinite product:
σ = ∏ k = 1 ∞ k 1 / 2 k = 1 1 / 2 2 1 / 4 3 1 / 8 4 1 / 16 … {\displaystyle \sigma =\prod _{k=1}^{\infty }k^{1/2^{k}}=1^{1/2}\;2^{1/4}\;3^{1/8}\;4^{1/16}\dots }
This can be easily rewritten into the far more quickly converging product representation
σ = ( 2 1 ) 1 / 2 ( 3 2 ) 1 / 4 ( 4 3 ) 1 / 8 ( 5 4 ) 1 / 16 … {\displaystyle \sigma =\left({\frac {2}{1}}\right)^{1/2}\left({\frac {3}{2}}\right)^{1/4}\left({\frac {4}{3}}\right)^{1/8}\left({\frac {5}{4}}\right)^{1/16}\dots }
which can then be compactly represented in infinite product form by:
σ = ∏ k = 1 ∞ ( 1 + 1 k ) 1 / 2 k {\displaystyle \sigma =\prod _{k=1}^{\infty }\left(1+{\frac {1}{k}}\right)^{1/2^{k}}}
Another product representation is given by:
σ = ∏ n = 1 ∞ ∏ k = 0 n ( k + 1 ) ( − 1 ) k + n ( n k ) {\displaystyle \sigma =\prod _{n=1}^{\infty }\prod _{k=0}^{n}(k+1)^{(-1)^{k+n}{\binom {n}{k}}}}
Expressions for ln σ {\displaystyle \ln \sigma } (sequence A114124 in the OEIS) include:
ln σ = ∑ k = 1 ∞ ln k 2 k {\displaystyle \ln \sigma =\sum _{k=1}^{\infty }{\frac {\ln k}{2^{k}}}}
ln σ = ∑ k = 1 ∞ ( − 1 ) k + 1 k Li k ( 1 2 ) {\displaystyle \ln \sigma =\sum _{k=1}^{\infty }{\frac {(-1)^{k+1}}{k}}{\text{Li}}_{k}\left({\tfrac {1}{2}}\right)}
ln σ 2 = ∑ k = 1 ∞ 1 2 k ( ln ( 1 + 1 k ) − 1 k ) {\displaystyle \ln {\frac {\sigma }{2}}=\sum _{k=1}^{\infty }{\frac {1}{2^{k}}}\left(\ln \left(1+{\frac {1}{k}}\right)-{\frac {1}{k}}\right)}
Integrals Integrals for ln σ {\displaystyle \ln \sigma } are given by:
ln σ = ∫ 0 1 1 − x ( x − 2 ) ln x d x {\displaystyle \ln \sigma =\int _{0}^{1}{\frac {1-x}{(x-2)\ln x}}dx}
ln σ = ∫ 0 1 ∫ 0 1 − x ( 2 − x y ) ln ( x y ) d x d y {\displaystyle \ln \sigma =\int _{0}^{1}\int _{0}^{1}{\frac {-x}{(2-xy)\ln(xy)}}dxdy}
Other formulas The constant σ {\displaystyle \sigma } arises when studying the asymptotic behaviour of the sequence
g 0 = 1 {\displaystyle g_{0}=1}
g n = n g n − 1 2 , n ≥ 1 {\displaystyle g_{n}=ng_{n-1}^{2},\qquad n\geq 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:
g n ∼ σ 2 n ( n + 2 − n − 1 + 4 n − 2 − 21 n − 3 + 138 n − 4 + O ( n − 5 ) ) − 1 {\displaystyle g_{n}\sim {\sigma ^{2^{n}}}\left(n+2-n^{-1}+4n^{-2}-21n^{-3}+138n^{-4}+O(n^{-5})\right)^{-1}}
Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent Φ ( z , s , q ) {\displaystyle \Phi (z,s,q)} :
ln σ = − 1 2 ∂ Φ ∂ s ( 1 / 2 , 0 , 1 ) {\displaystyle \ln \sigma =-{\frac {1}{2}}{\frac {\partial \Phi }{\partial s}}\!\left(1/2,0,1\right)}
If one defines the Euler-constant function (which gives Euler's constant for z = 1 {\displaystyle z=1} ) as:
γ ( z ) = ∑ n = 1 ∞ z n − 1 ( 1 n − ln ( n + 1 n ) ) {\displaystyle \gamma (z)=\sum _{n=1}^{\infty }z^{n-1}\left({\frac {1}{n}}-\ln \left({\frac {n+1}{n}}\right)\right)}
one has:
γ ( 1 2 ) = 2 ln 2 σ {\displaystyle \gamma ({\tfrac {1}{2}})=2\ln {\frac {2}{\sigma }}}
Universality One may define a "continued binary expansion" for all real numbers in the set ( 0 , 1 ] {\displaystyle (0,1]} , similarly to the decimal expansion or simple continued fraction expansion. This is done by considering the unique base-2 representation for a number x ∈ ( 0 , 1 ] {\displaystyle x\in (0,1]} which does not contain an infinite tail of 0's (for example write one half as 0.01111... 2 {\displaystyle 0.01111..._{2}} instead of 0.1 2 {\displaystyle 0.1_{2}} ). Then define a sequence ( a k ) ⊆ N {\displaystyle (a_{k})\subseteq \mathbb {N} } which gives the difference in positions of the 1's in this base-2 representation. This expansion for x {\displaystyle x} is now given by:
x = ⟨ a 1 , a 2 , a 3 , . . . ⟩ {\displaystyle x=\langle a_{1},a_{2},a_{3},...\rangle }
For example the fractional part of Pi we have:
{ π } = 0.14159 26535 89793... = 0.00100 10000 11111... 2 {\displaystyle \{\pi \}=0.14159\,26535\,89793...=0.00100\,10000\,11111..._{2}} (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:
π − 3 = ⟨ 3 , 3 , 5 , 1 , 1 , 1 , 1... ⟩ {\displaystyle \pi -3=\langle 3,3,5,1,1,1,1...\rangle } (sequence A320298 in the OEIS) This gives a bijective map ( 0 , 1 ] ↦ N N {\displaystyle (0,1]\mapsto \mathbb {N} ^{\mathbb {N} }} , such that for every real number x ∈ ( 0 , 1 ] {\displaystyle x\in (0,1]} we uniquely can give:
x = ⟨ a 1 , a 2 , a 3 , . . . ⟩ :⇔ x = ∑ k = 1 ∞ 2 − ( a 1 + . . . + a k ) {\displaystyle x=\langle a_{1},a_{2},a_{3},...\rangle :\Leftrightarrow x=\sum _{k=1}^{\infty }2^{-(a_{1}+...+a_{k})}}
It can now be proven that for almost all numbers x ∈ ( 0 , 1 ] {\displaystyle x\in (0,1]} the limit of the geometric mean of the terms a k {\displaystyle a_{k}} converges to Somos' constant. That is, for almost all numbers in that interval we have:
σ = lim n → ∞ a 1 a 2 . . . a n n {\displaystyle \sigma =\lim _{n\to \infty }{\sqrt[{n}]{a_{1}a_{2}...a_{n}}}}
Somos' constant is universal for the "continued binary expansion" of numbers x ∈ ( 0 , 1 ] {\displaystyle x\in (0,1]} in the same sense that Khinchin's constant is universal for the simple continued fraction expansions of numbers x ∈ R {\displaystyle x\in \mathbb {R} } .
Generalizations The generalized Somos' constants may be given by:
σ t = ∏ k = 1 ∞ k 1 / t k = 1 1 / t 2 1 / t 2 3 1 / t 3 4 1 / t 4 … {\displaystyle \sigma _{t}=\prod _{k=1}^{\infty }k^{1/t^{k}}=1^{1/t}\;2^{1/t^{2}}\;3^{1/t^{3}}\;4^{1/t^{4}}\dots }
for t > 1 {\displaystyle t>1} . The following series holds:
ln σ t = ∑ k = 1 ∞ ln k t k {\displaystyle \ln \sigma _{t}=\sum _{k=1}^{\infty }{\frac {\ln k}{t^{k}}}}
We also have a connection to the Euler-constant function:
γ ( 1 t ) = t ln ( t ( t − 1 ) σ t t − 1 ) {\displaystyle \gamma ({\tfrac {1}{t}})=t\ln \left({\frac {t}{(t-1)\sigma _{t}^{t-1}}}\right)}
and the following limit, where γ {\displaystyle \gamma } is Euler's constant:
lim t → 0 + t σ t + 1 t = e − γ {\displaystyle \lim _{t\to 0^{+}}t\sigma _{t+1}^{t}=e^{-\gamma }}
See also Euler's constant Khinchin's constant Binary number Ergodic theory List of mathematical constants
References