Definition:Landau-Ramanujan Constant

From ProofWiki
Jump to navigation Jump to search

Definition

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

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

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


Also see


Source of Name

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


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)