Asymptotic distribution of integral points on the three-dimensional sphere

Abstract

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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The linking number in systems with Periodic Boundary Conditions

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Estimates of Petersson's inner squares of cusp forms and arithmetic applications

“…In th~s section we formulate an analogue of Theorem 7 for ellipsoids with an odd number l _> 3 of variables. The method of proof generalizes the method of [20] and it is based on considerations from [19] and on Iwaniec's estimates for the Fourier coefficients of modular forms of half-integral weight l/2 > 5/2 (see [21]). THEOREM 9.…”