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, stating that for large $x$, the number of positive integers below $x$ that are the sum of two square numbers behaves asymptotically as

:$\dfrac.$

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

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 $N(x)$ is the number of positive integers less than $x$ that are the sum of two squares, then
:$\lim_\ \left(\dfrac\right)=b\approx 0.764223653589220662990698731250092328116790541$ ,
where $b$ is the Landau–Ramanujan constant.

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

$b = \frac\prod_ \left(1 - \frac\right)^ = \frac \prod_ \left(1 - \frac\right)^.$

History

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

References

{{DEFAULTSORT:Landau-Ramanujan constant
Additive number theory
Analytic number theory
Mathematical constants
Srinivasa Ramanujan