Sierpiński constant
The Sierpiński constant is a mathematical constant named after the Polish mathematician Wacław Sierpiński . It can be defined by the following expression, among others:
where the number of representations of in the form with integers and taking the sequence into account is the circle number and the natural logarithm .
Forms of representation
An explicit expression for the Sierpiński constant is
with the Euler-Mascheroni constant and the gamma function . Because of the relation
the alternative representation results
The decimal expansion of is
r n (k) function
0 | 1 |
1 | 4th |
2 | 4th |
3 | 0 |
4th | 4th |
5 | 8th |
6th | 0 |
7th | 0 |
... | ... |
25th | 12 |
... | ... |
65 | 16 |
... | ... |
The Sierpiński constant occurs when examining the asymptotics of the function (referred to in English as the sum of squares )
for the case (the case is about the case in Jacobi's theorem ).
For example, = 0, because the number 3 cannot be represented as the sum of two square numbers, while = 8, because 13 can be formed as the sum of the square numbers 9 and 4 in two different sequences and , each in four constellations of signs.
literature
- Wacław Sierpiński : O sumowaniu szeregu , gdzie τ (n) oznacza liczbę rozkładów liczby n na sumę kwadratów dwóch liczb całkowitych (About the summation of the series where τ ( n ) denotes the number of representations of n as the sum of two squares), Prace matematyczno-fizyczne 18, 1907, pp. 1–60 (Polish; in the Internet archive ; " K = 2.5849817596" on p. 27 ; yearbook report )
- Steven R. Finch: Sierpinski's constant , Chapter 2.10 in Mathematical constants , Cambridge University Press, Cambridge 2003, ISBN 0-521-81805-2 , pp. 122–125 (English; Finch's website for the book with errata and addenda: Mathematical Constants . )
Web links
- Eric W. Weisstein : Sierpiński Constant . In: MathWorld (English).
- Eric W. Weisstein : Sum of Squares Function . In: MathWorld (English).
- Sequence A062083 in OEIS ( continued fraction expansion of K )
- Episode A108905 in OEIS (Engel development from K )