In mathematics, the Landau-Ramanujan constant occurs in a number theory result that the proportion of positive integers less than x which are the sum of two square numbers is, for large x, roughly proportional to

<math>1/{\sqrt{\ln(x)}}.<math>

The constant of proportionality is the Landau-Ramanujan constant.

More formally, if N(x) is the number of positive integers less than x which are the sum of two squares, then

<math>\lim_{x\rightarrow\infty} \frac{N(x)\sqrt{\ln(x)}}{x}\approx 0.76422365358922066299069873125.<math>