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 formula_1, the number of positive integers below formula_1 that are the sum of two square numbers behaves asymptotically as This constant "b" was rediscovered in 1913 by Srinivasa Ramanujan, in the first letter he wrote to G.H. Hardy. 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, 3 × 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 formula_4 is the number of positive integers less than formula_1 that are the sum of two squares, then where formula_7 is the Landau–Ramanujan constant. History. This constant was  stated by Landau in the limit form above; Ramanujan instead approximated formula_4 as an integral, with the same constant of proportionality, and with a slowly growing error term.