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
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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. Landau: Über 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)
- 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $0,764 \ldots$
- Weisstein, Eric W. "Landau-Ramanujan Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Landau-RamanujanConstant.html