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]