Number of Natural Numbers Less than x which are Squares or Sums of Two Squares

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Theorem

Let $x$ be a real number.

The number of natural numbers smaller than $x$ which are either square or the sum of $2$ squares is given by the expression:

$\map N x \approx \dfrac {k x} {\sqrt {\ln x} }$

where $k$ is given by:

$k = \sqrt {\dfrac 1 2 \ds \prod_{\substack {r \mathop = 4 n \mathop + 3 \\ \text {$r$ prime} } } \paren {1 - \dfrac 1 {r^2} }^{-1} }$

The number $k$ is known as the Landau-Ramanujan constant:

\(\ds k\) \(=\) \(\ds \sqrt {\dfrac 1 2 \ds \prod_{\substack {r \mathop = 4 n \mathop + 3 \\ \text {$r$ prime} } } \paren {1 - \dfrac 1 {r^2} }^{-1} }\)
\(\ds \) \(\approx\) \(\ds 0 \cdotp 76422 \, 3653 \ldots\)


Proof




Historical Note

This theorem was demonstrated by Edmund Georg Hermann Landau in $1908$.

Srinivasa Aiyangar Ramanujan re-stated the theorem in a slightly different form.


Sources

  • 1908: E. LandauÜber die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate (Arch. Math. Phys Vol. 13: pp. 305 – 312)