Distribution of arithmetical functions. A survey
Let two lattices have the same number of points on each hyperbolic surface . We investigate the case when Λ’, Λ” are sublattices of of the same prime index and show that then Λ’ and Λ” must coincide up to renumbering the coordinate axes and changing their directions.
We discuss the distribution of Mordell-Weil ranks of the family of elliptic curves y² = (x + αf²)(x + βbg²)(x + γh²) where f,g,h are coprime polynomials that parametrize the projective smooth conic a² + b² = c² and α,β,γ are elements from ℚ̅. In our previous papers we discussed certain special cases of this problem and in this article we complete the picture by proving the general results.
In this article we study, using elementary and combinatorial methods, the distribution of quadratic non-residues which are not primitive roots modulo or for an odd prime and an integer.