Let ~(X)= ~+ xz+x ~z z) X=(~4,~zx3) ; ~(~) let r(n) be the number of integral solutions of the equation QCX) = (1)The following theorem is proved: let n = I, 2, 3, 5, 6 (mod 8) and let r(n, ~) be the number of integral solutions of equation (i) such that Y = ~/~ E ~ where ~ is an arbitrary convex domain with a piecewise smooth boundary on the unit sphere S: Q(Y) = i. Then where D(~) is a measure, normalized by the condition ~(S) = i. A similar result is obtained for the three-dimensional ellipsoid of general form. The mentioned theorem, in combination with the classical Guass--Siegel results on r(n), yields the uniform distribution of the integral points on the three-dimensional sphere (i). i. The questions of the representability of integers by positive definite ternary quadratic forms and the closely related questions on the number of integral points in domains on ellipsoids have been investigated by a series of authors. Ternary quadratic forms of general form have been considered for the first time by Yu. V, Linnik (the discrete ergodic method, going back to his 1939-1940 investigations (see [i]); for applications to ellipsoids, the ergodic method has been developed by A. V. Malyshev [2] and Yu. G. Teterin [3], [4]. Another method of investigation of the ellipsoids belongs to E. P. Golubeva [5], [6]. For their use, both these methods require some assumptions, for example, unproved although weak conjectures on the zeros of Dirichlet L-series, and so far they do not give the final solution of the indicated problems.We consider the most important case, namely the three-dimensional sphere Sph3 ; ~ + ~f + ~ = ~ .

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This paper borders on []1], where a connection is established between an analogue of Artin's conjecture on the primitive roots for the case of a quadratic field and the problem regarding the number of classes of binary indefinite quadratic forms, whose discriminants have an increasing quadratic part.The presence of an additional multiplicative structure makes this problem simpler than the problem of the upper estimation of the number of classes in the general case.Thus, a direct generalization of Theorem 1 of [1] (see also Lemma 2 of the present paper) leads to the estimate e~.zwhere d ~ m 2, d > 0, is a fixed number, p is a prime number, while h(m) is the number of classes of binary quadratic forms of determinant m. We mention that a trivial estimate of this sum is Od(x~'), while the estimate Od(Xl+'), consistent with the various conjectures on the behavior of the averaged value of the number of classes over all determinants (see, for example, [2]), to within e, is obviously unimprovable. The proof of (1) is analogous to the proof of Chebyshev's estimate for ~r(x). However, in our case, unlike the classical one, these considerations lead to an estimate that is not proper with respect to order.In addition, it has been shown in [1] that the transfer of Hooley's results [3] on the Artin conjecture to the case of quadratic fields allows us to conclude that, under the assumption of the validity of the Riemann hypothesis for the corresponding fields the relation h(dp 2) = 2h(d) (and, in particular, h(dp 2) ---2) is satisfied for a subsequence of p's of positive density. (We note that h(dp 2) __. 2 for p $ ~ ~ since for such determinants the class group splits into at least two genera).In this paper we show that the progress in the proof of Artin's conjecture, achieved recently in [4], [5], has a counterpart in the considered problem. A small modification of the arguments of these works allows us to prove (without using Riemann's hypothesis) that the relation h(dp ~) --2 is satisfied sufficiently often (see Theorem 1 below). We note that the numbers 7, 11, and 19, occurring in the formulation of Theorem 1, can be replaced by any three fixed prime numbers q, for each of which h(q) = 1. We mention that in the proof we follow Heath-Brown [5]; however, in order to obtain the corollaries of Theorem 1, which have a more natural formulation, it would be sufficient to carry over the arguments from [4] to our case.

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