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 is the number of positive integers less than that are the sum of two squares, then
The Landau-Ramanujan constant can also be written as an infinite product:
History
This constant was stated by Landau in the limit form above; Ramanujan instead approximated as an integral, with the same constant of proportionality, and with a slowly growing error term.[3]
References
^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
^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.