# 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.