Landau–Ramanujan constant

In mathematics and the field of number theory, the Landau–Ramanujan constant is a number that occurs in a theorem stating that for large x, the number of positive integers below x that are the sum of two square numbers varies as

${\dfrac {x}{\sqrt {\ln(x)}}}.$

The constant is named after its discoverer, Srinivasa Ramanujan.

Definition

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 the 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.

If N(x) is the number of positive integers less than x that are the sum of two squares, then

$\lim _{x\rightarrow \infty }\ N(x)\left/{\dfrac {x}{\sqrt {\ln(x)}}}\right.\approx 0.764223653589220662990698731250092328116790541$ (sequence A064533 in the OEIS).

The number appearing on the right hand side of this formula is the Landau–Ramanujan constant.

History

This constant was discovered independently by Srinivasa Ramanujan and Edmund Landau. Landau stated it 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.