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Lattice points in some special three-dimensional convex bodies with points of Gaussian curvature zero at the boundary

Ekkehard Krätzel (2002)

Commentationes Mathematicae Universitatis Carolinae

We investigate the number of lattice points in special three-dimensional convex bodies. They are called convex bodies of pseudo revolution, because we have in one special case a body of revolution and in another case even a super sphere. These bodies have lines at the boundary, where all points have Gaussian curvature zero. We consider the influence of these points to the lattice rest in the asymptotic representation of the number of lattice points.

Lattice points in super spheres

Ekkehard Krätzel (1999)

Commentationes Mathematicae Universitatis Carolinae

In this article we consider the number R k , p ( x ) of lattice points in p -dimensional super spheres with even power k ≥ 4 . We give an asymptotic expansion of the d -fold anti-derivative of R k , p ( x ) for sufficiently large d . From this we deduce a new estimation for the error term in the asymptotic representation of R k , p ( x ) for p < k < 2 p - 4 .

Linear congruences and a conjecture of Bibak

Chinnakonda Gnanamoorthy Karthick Babu, Ranjan Bera, Balasubramanian Sury (2024)

Czechoslovak Mathematical Journal

We address three questions posed by K. Bibak (2020), and generalize some results of K. Bibak, D. N. Lehmer and K. G. Ramanathan on solutions of linear congruences ∑ i = 1 k a i x i ≡ b ( mod n ) . In particular, we obtain explicit expressions for the number of solutions, where x i ’s are squares modulo n . In addition, we obtain expressions for the number of solutions with order restrictions x 1 ≥ ⋯ ≥ x k or with strict order restrictions x 1 > ⋯ > x k in some special cases. In these results, the expressions for the number of solutions involve Ramanujan...

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