Sierpiński constant

This article deals with the Sierpinski constant . For the sequence of numbers named after Sierpinski, see the Sierpiński number . ${\displaystyle K}$

The Sierpiński constant is a mathematical constant named after the Polish mathematician Wacław Sierpiński . It can be defined by the following expression, among others:

${\displaystyle K=\lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {r_{2}(k)}{k}}-\pi \ln n\right),}$

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 . ${\displaystyle r_{2}(k)}$ ${\displaystyle k}$ ${\displaystyle a^{2}+b^{2}}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \pi }$ ${\displaystyle \ln }$

Forms of representation

An explicit expression for the Sierpiński constant is ${\displaystyle K}$

${\displaystyle K=\pi \left(2\gamma +\ln {\frac {4\pi ^{3}}{\Gamma (1/4)^{4}}}\right)}$

with the Euler-Mascheroni constant and the gamma function . Because of the relation ${\displaystyle \gamma }$ ${\displaystyle \Gamma }$

${\displaystyle \Gamma (1/4)={\frac {\pi {\sqrt {2}}}{\Gamma (3/4)}}}$

the alternative representation results

${\displaystyle K=\pi \left(2\gamma +4\ln \Gamma \left({\frac {3}{4}}\right)-\ln \pi \right).}$

The decimal expansion of is ${\displaystyle K}$

${\displaystyle K=2{,}58498\ 17595\ 79253\ 21706\ 58935\ 87383\ 17116\ 00880\ 51651\ 85263\ \dots }$(Follow A062089 in OEIS )

r _n (k) function

${\displaystyle k}$	${\displaystyle r_{2}(k)}$
0	1
1	4th
2	4th
3	0
4th	4th
5	8th
6th	0
7th	0
...	...
25th	12
...	...
65	16
...	...

(Follow A004018 in OEIS ).

The Sierpiński constant occurs when examining the asymptotics of the function (referred to in English as the sum of squares )

${\displaystyle r_{n}(k)={\bigl |}{\bigl \{}(a_{1},a_{2},\ldots ,a_{n})\in \mathbb {Z} ^{n}\mid a_{1}^{2}+a_{2}^{2}+\ldots +a_{n}^{2}=k{\bigr \}}{\bigr |}}$

for the case (the case is about the case in Jacobi's theorem ). ${\displaystyle n=2}$ ${\displaystyle n=4}$

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. ${\displaystyle r_{2}(3)}$ ${\displaystyle r_{2}(13)}$ ${\displaystyle (\pm 3)^{2}+(\pm 2)^{2}}$ ${\displaystyle (\pm 2)^{2}+(\pm 3)^{2}}$

literature

• Wacław Sierpiński : O sumowaniu szeregu , gdzie τ (n) oznacza liczbę rozkładów liczby n na sumę kwadratów dwóch liczb całkowitych ${\displaystyle \textstyle \sum _{n>a}^{n\leq b}\tau (n)f(n)}$ (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 ) ${\displaystyle \textstyle \sum _{n>a}^{n\leq b}\tau (n)f(n)}$
• 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 )