Definition:Landau-Ramanujan Constant

Definition

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

\(\ds k\)

\(=\)

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

\(\ds \)

\(\approx\)

\(\ds 0 \cdotp 76422 \, 3653 \ldots\)

This sequence is A064533 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Also see

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

Source of Name

This entry was named for Edmund Georg Hermann Landau and Srinivasa Aiyangar Ramanujan.

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)

• Weisstein, Eric W. "Landau-Ramanujan Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Landau-RamanujanConstant.html