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
x
x
\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]
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)
x
\limx → infty \left(\dfrac{N(x)}{\dfrac{x}{\sqrt{log(x)}}}\right)=b ≈ 0.764223653589220662990698731250092328116790541
b
The Landau-Ramanujan constant can also be written as an infinite product:
\begin{align}b&=
1 | |
\sqrt{2 |
This constant was stated by Landau in the limit form above; Ramanujan instead approximated
N(x)