Landau-Ramanujan constant

The Landau-Ramanujan constant is one of the mathematical constants and as such belongs in number theory . Its name refers to the two important mathematicians Edmund Landau and Srinivasa Ramanujan , who independently proved their existence. The Landau-Ramanujan constant is denoted by and has approximately the representation of a decimal number ${\displaystyle K}$ ${\displaystyle K=0{,}7642236535892206\ldots }$

The investigation of the Landau-Ramanujan constants is related to the question of which natural numbers can be represented as the sum of two square numbers and the resulting problem of asymptotically determining the proportion of these numbers in the natural numbers .

Formulas

For a positive real number, be the number of natural numbers that can be represented as the sum of two square numbers. Landau and Ramanujan independently proved that is asymptotically proportional to , i.e. that is, the limit value exists ${\displaystyle B(x)}$ ${\displaystyle x}$ ${\displaystyle n\leq x}$ ${\displaystyle B(x)}$ ${\displaystyle {\frac {x}{\sqrt {\ln(x)}}}}$

(I) , ${\displaystyle K=\lim _{x\rightarrow \infty }{\frac {B(x)}{x}}{\sqrt {\ln(x)}}}$

where stands for the natural logarithm of . The limit value is called the Landau-Ramanujan constant . ${\displaystyle \ln(x)}$ ${\displaystyle x}$ ${\displaystyle K}$

It also applies:

(II) ${\displaystyle {\begin{aligned}K&={\frac {1}{\sqrt {2}}}\cdot \prod \limits _{p\;{\text{Primzahl mit}} \atop \;\ p\equiv 3\;({\text{mod}}\;4)}{\bigl (}1-{\frac {1}{p^{2}}}{\bigr )}^{-{\frac {1}{2}}}\\&={\frac {\pi }{4}}\cdot \prod \limits _{p\;{\text{Primzahl mit}} \atop \;\ p\equiv 1\;({\text{mod}}\;4)}{\bigl (}1-{\frac {1}{p^{2}}}{\bigr )}^{\frac {1}{2}}\\\end{aligned}}}$

In addition, there are other formulas that relate the Landau-Ramanujan constant to, for example, the Riemann zeta function , the Dirichlet beta function , the Euler-Mascheroni constant and the lemniscate constants .

Derivation of the second equation in II

The second equation at II results from the Euler product representation of the Riemann zeta function on the half plane . Because from it follows for   with the help of a well-known number formula   of analysis : ${\displaystyle \zeta (z)}$ ${\displaystyle \operatorname {Re} (z)>1}$ ${\displaystyle z=2}$

${\displaystyle {\begin{aligned}{\frac {{\pi }^{2}}{6}}&={\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}}\\&=\zeta (2)\\&={\prod \limits _{p\;{\text{Primzahl}}}{\bigl (}1-{\frac {1}{p^{2}}}{\bigr )}^{-1}}\\&={\frac {4}{3}}\cdot A\cdot B\\\end{aligned}}}$

With

${\displaystyle {\begin{aligned}A&=\prod \limits _{p\;{\text{Primzahl mit}} \atop \;\ p\equiv 1\;({\text{mod}}\;4)}{\bigl (}1-{\frac {1}{p^{2}}}{\bigr )}^{-1}\\\end{aligned}}}$

and

${\displaystyle {\begin{aligned}B&=\prod \limits _{p\;{\text{Primzahl mit}} \atop \;\ p\equiv 3\;({\text{mod}}\;4)}{\bigl (}1-{\frac {1}{p^{2}}}{\bigr )}^{-1}\\\end{aligned}}}$

The last equation of the above chain of equations goes into the fact that a prime number is either equal to 2 or odd and in the latter case modulo 4 either has the remainder 1 or 3.

So it arises

${\displaystyle {\begin{aligned}{\frac {1}{2}}\cdot B&={\frac {{\pi }^{2}}{16\cdot A}}\\\end{aligned}}}$

and thus

${\displaystyle {\begin{aligned}{\frac {1}{\sqrt {2}}}\cdot B^{\frac {1}{2}}&={{\bigl (}{\frac {1}{2}}\cdot B{\bigr )}}^{\frac {1}{2}}\\&={\frac {\pi }{4\cdot A^{\frac {1}{2}}}}={\frac {\pi }{4}}\cdot {A^{-{\frac {1}{2}}}}\\\end{aligned}}}$ 

and finally the equation to be shown.

See also

• Mathematical constant
• List of special numbers

literature

• Steven R. Finch: Mathematical Constants (=  Encyclopedia of Mathematics and its Applications . Volume 94 ). Cambridge University Press , Cambridge [u. a.] 2003, ISBN 0-521-81805-2 ( MR2003519 ).
• Siegfried Gottwald (ed.): Lexicon of important mathematicians . Verlag Harri Deutsch, Thun 1990, ISBN 3-8171-1164-9 .
• Konrad Knopp : Theory and Application of the Infinite Series (=  The Basic Teachings of the Mathematical Sciences in Individual Representations . Volume 2 ). 5th, corrected edition. Springer-Verlag, Berlin [a. a.] 1964, ISBN 3-540-03138-3 .
• Daniel Shanks : The Second-Order Term in the Asymptotic Expansion of B (x) . In: Mathematics of Computation . tape 18 , 1964, pp. 75-86 , JSTOR : 2003407 .

Web links

• Eric W. Weisstein : Landau-Ramanujan Constant . In: MathWorld (English).

Individual evidence

1. ↑ Steven R. Finch: Mathematical Constants (=  Encyclopedia of Mathematics and its Applications . Volume 94 ). Cambridge Univity Press, Cambridge [et. a.] 2003, ISBN 0-521-81805-2 , pp. 98-104 ( MR2003519 ).
2. ↑ Follow A064533 in OEIS
3. ↑ E. Landau: On the division of positive integers into four classes according to the minimum number of squares required for their additive composition . In: Arch. Math. Phys. , 13, 1908, pp. 305-312.
4. ↑ Konrad Knopp : Theory and Application of the Infinite Series (=  The Basic Teachings of Mathematical Sciences in Individual Representations . Volume 2 ). 5th, corrected edition. Springer-Verlag, Berlin [a. a.] 1964, ISBN 3-540-03138-3 , pp. 461 .