Geodesic
Prime vectors in degenerate lattices
Sbornik. Mathematics, Tome 54 (1986) no. 2, pp. 279-295 Cet article a éte moissonné depuis la source Math-Net.Ru

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In this paper the distribution of prime vectors (i.e., vectors with prime components) in degenerate lattices $AZ^n+\overline b$ is investigated, and asymptotic formulas are obtained for the fraction $\pi(N,AZ^n+\overline b)$ which are valid under certain restrictions on the matrix $A$, where $A\in Z^{m\times n}$, $\overline b\in Z^m$, and $\pi(N,AZ^n+\overline b)$ is the number of prime vectors of the degenerate lattice $AZ^n+\overline b$ with components not exceeding $N$. The main idea is to reduce the problem to that of solving systems of linear algebraic equations in prime numbers belonging to given arithmetic progressions. An asymptotic formula for the number of solutions of such systems is calculated with the help of a multidimensional variant of the circle method. Bibliography: 12 titles. 
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M. I. Tulyaganova. Prime vectors in degenerate lattices. Sbornik. Mathematics, Tome 54 (1986) no. 2, pp. 279-295. http://geodesic.mathdoc.fr/item/SM_1986_54_2_a0/

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