On Sums of Two Coprime k-th Powers.
On sums of two cubes: an Ω₊-estimate for the error term
The arithmetic function counts the number of ways to write a natural number n as a sum of two kth powers (k ≥ 2 fixed). The investigation of the asymptotic behaviour of the Dirichlet summatory function of leads in a natural way to a certain error term which is known to be in mean-square. In this article it is proved that as t → ∞. Furthermore, it is shown that a similar result would be true for every fixed k > 3 provided that a certain set of algebraic numbers contains a sufficiently...
On sums of two kth powers: a mean-square asymptotics over short intervals
On sums of two kth powers: an asymptotic formula for the mean square of the error term
On sums over primes
On sum-sets and product-sets of complex numbers
We give a simple argument that for any finite set of complex numbers , the size of the the sum-set, , or the product-set, , is always large.
On sumsets and spectral gaps
On S-unit equations in two unknowns.
On supercharacter theoretic generalizations of monomial groups and Artin's conjecture
We extend the notions of quasi-monomial groups and almost monomial groups in the framework of supercharacter theories, and we study their connection with Artin’s conjecture regarding the holomorphy of Artin -functions.
On superpseudoprimes
On Sylow 2-subgroups of K2OF for quadratic number fields F.
On symmetric and antisymmetric balanced binary sequences.
On symmetric and unsymmetric theta functions over a real quadratic field
On symmetric square values of quadratic polynomials
On systems of composite Lehmer numbers with prime indices
On systems of diophantine equations with a large number of solutions
We consider systems of equations of the form and , which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.
On systems of linear inequalities
We show in detail that the category of general Roth systems or the category of semi-stable systems of linear inequalities of slope zero is a neutral Tannakian category. On the way, we present a new proof of the semi-stability of the tensor product of semi-stable systems. The proof is based on a numerical criterion for a system of linear inequalities to be semi-stable.
On systems of three quadratic forms
On Tate’s refinement for a conjecture of Gross and its generalization
We study Tate’s refinement for a conjecture of Gross on the values of abelian -function at and formulate its generalization to arbitrary cyclic extensions. We prove that our generalized conjecture is true in the case of number fields. This in particular implies that Tate’s refinement is true for any number field.
On Tate-Shafarevich groups of .