Landau-Ramanujan Constant

Let B(x) denote the number of positive integers not exceeding x which can be expressed as a sum of two squares. E. Landau and S. Ramanujan independently proved that:

where K is given by:

Landau further proved that

where C is the constant

L=2.622057554... is Gauss' lemniscate constant and is Euler's constant. D. Shanks computed C=0.581948659... to 10 digits. See the references for improvements.

In the Inverse Symbolic Calculator web pages, Simon Plouffe gave a highly accurate approximation of K as well as one for the second-order constant C.

See also Moree and Cazaran's survey on the subject.

Acknowledgements

I'm grateful to Allen MacLeod, Dave Hare, Philippe Flajolet, Paul Zimmermann, Victor Adamchik, Jeffrey Golden and William Gosper for their correspondence.

More details and references (contact Steven Finch).
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