Landau–Ramanujan constant

In mathematics and the field of number theory, the Landau–Ramanujan constant is the positive real number b that occurs in a theorem proved by Edmund Landau in 1908,^[1] stating that for large ${\displaystyle x}$, the number of positive integers below ${\displaystyle x}$ that are the sum of two square numbers behaves asymptotically as

${\displaystyle {\dfrac {bx}{\sqrt {\log(x)}}}.}$

This constant b was rediscovered in 1913 by Srinivasa Ramanujan, in the first letter he wrote to G.H. Hardy.^[2]

Sums of two squares

By the sum of two squares theorem, the numbers that can be expressed as a sum of two squares of integers are the ones for which each prime number congruent to 3 mod 4 appears with an even exponent in their prime factorization. For instance, 45 = 9 + 36 is a sum of two squares; in its prime factorization, 3² × 5, the prime 3 appears with an even exponent, and the prime 5 is congruent to 1 mod 4, so its exponent can be odd.

Landau's theorem states that if ${\displaystyle N(x)}$ is the number of positive integers less than ${\displaystyle x}$ that are the sum of two squares, then

${\displaystyle \lim _{x\rightarrow \infty }\ \left({\dfrac {N(x)}{\dfrac {x}{\sqrt {\log(x)}}}}\right)=b\approx 0.764223653589220662990698731250092328116790541}$ (sequence A064533 in the OEIS),

where ${\displaystyle b}$ is the Landau–Ramanujan constant.

The Landau-Ramanujan constant can also be written as an infinite product:

${\displaystyle b={\frac {1}{\sqrt {2}}}\prod _{p\equiv 3{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{-1/2}={\frac {\pi }{4}}\prod _{p\equiv 1{\pmod {4}}}\left(1-{\frac {1}{p^{2}}}\right)^{1/2}.}$

History

This constant was stated by Landau in the limit form above; Ramanujan instead approximated ${\displaystyle N(x)}$ as an integral, with the same constant of proportionality, and with a slowly growing error term.^[3]

References

1. Edmund Landau, Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate, Archiv der Mathematik und Physik (3) 13 (1908), 305-312
2. S. Ramanujan, letter to G.H. Hardy, 16 January, 1913; see: P. Moree and J. Cazaran, On a claim of Ramanujan in his first letter to Hardy, Exposition. Math. 17 (1999), no.4, 289-311.
3. Weisstein, Eric W. "Landau–Ramanujan Constant". MathWorld.