Computing the Landau-Ramanujan Constant

Let denote the number of positive integers not exceeding which can be expressed as a sum of two squares excepting 1. The function clearly approaches infinity as increases, but one can ask how fast it does so. Landau and Ramanujan both independently proved that the following limit exists:

It is straightforward to estimate this limit directly using Mathematica.

This computes values of [x].

Here is a plot of successive approximations to the limit.

In 1996 Philippe Flajolet and Ilan Vardi deduced the following exact formula for the limit:

They computed 1024 digits in the decimal expansion of this limit. Using essentially the following direct Mathematica program, Victor Adamchik was then able to calculate 5100 digits.

On a typical computer, this takes just a few seconds.

This typically takes a minute or so.

This takes a few minutes.

This takes a few hours.

Since Mathematica has no built-in limits, waiting longer on a larger computer will always give higher precision...