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Acta Arith algebraic integers algebraic number field algebraic numbers apply assume class number completes the proof computation conjecture conjugate Conjugate algebraic contains converges Corollary covering systems defined definite Dirichlet L-functions discriminant divisor Eichler order element equation estimate exists finite prime follows formula gives Hence hypothesis idealcomplex implies inequality infinite irreducible J. W. S. CASSELS Lemma lemme Math maximal lattices maximally integral lattice mesure minimal polynomial modulus q natural density non-zero norm notation number field number of primes number theory obtain polynomial positive integers prime ideals prime numbers proof of Theorem Proposition prove quadratic extension quaternary forms quaternary space quaternion algebra ramified rational integers real number residue classes result roots of unity satisfy Section similitude classes solutions square square-free square-free integer sufficiently Suppose system having property theory of numbers upper bound values zeta function Zp[x