Warren Dicks and J. Porti, Expressing a number as the sum of two coprime squares, Collect. Math. 49 (1998), 283-291.
 


Addenda

December 3, 1999


Ignasi Mundet of the Universidad Autónoma de Madrid gives an elegant elementary proof of our main result in [1]. 

Stefan Kühnlein of Universität Fridericiana (TH) Karlsruhe has sent us the following interesting comments. See also his Math Review of the paper.

The general formula for asymptotically expressing the number

#{(x_0,...,x_n) \in Z^{n+1} | x_0^2 + ... + x_n^2 < R^2,MCD{x_0,...,x_n}=1}

is

( pi^{(n+1)/2} / [(n+1) \Gamma((n+1)/2) \zeta(n+1)] ) * R^{n+1}.



This can be proved by Moebius-inversion and is related to estimating the number of points of height less than R in projective n-space over the rationals.

For n=1, this appears as the case of Theorem 459 of [2] where P is the ball of radius R. I don't know a reference for general n in the literature. I learned it in a talk of Jens Franke some years ago and it seems to be "well-known" to arithmetic geometers. In this setting, however, one is not interested in the congruence conditions.

In another direction, one can extend the main result to deal with congruences other than m|c: one can also impose the condition that d lies in some specified subgroup of the group of units of Z/mZ (e.g. m|c, m|(d-1)), and then the asymptotic expression is gotten from the index of the corresponding subgroup containing the congruence subgroup \Gamma_1(m).

The fact that the covolume of SL_2(Z) on the upper half plane is pi/3 can be viewed as one of the formulae relating covolumes of arithmetic groups on the associated symmetric space to (products of) special values of zeta functions.

References

[1] Ignasi Mundet i Riera, On the density of coprime pairs of numbers, preprint, 1999, 3 pages.
[2] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th edition, Clarendon Press, Oxford, 1979.

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