The Romanoff theorem revisited
The Rosser-Iwaniec sieve in number fields, with an application
The extensions of degree 6 with minimum discriminant.
The Schrödinger density and the Talbot effect
We study the local properties of the time-dependent probability density function for the free quantum particle in a box, i.e. the squared magnitude of the solution of the Cauchy initial value problem for the Schrödinger equation with zero potential, and the periodic initial data. √δ-families of initial functions are considered whose squared magnitudes approximate the periodic Dirac δ-function. The focus is on the set of rectilinear domains where the density has a special character, in particular,...
The Schur Multiplicator of SL(2, Z/mZ) and the Congruence Subgroup Property.
The Schur Subgroup of a Real Cyclotomic Field.
The second moment of Dirichlet twists of Hecke L-functions
The second moment of quadratic twists of modular L-functions
We study the second moment of the central values of quadratic twists of a modular -function. Unconditionally, we obtain a lower bound which matches the conjectured asymptotic formula, while on GRH we prove the asymptotic formula itself.
The second moment of the symmetric square -functions.
The Selberg trace formula for the Picard group SL(2, Z[i])
The Selberg zeta function and a local trace formula for Kleinian groups.
The Selberg zeta-function for cocompact discrete subgroups of PSL(2,C)
The Selberg-Delange method in short intervals with an application
We establish a general mean value result for arithmetic functions over short intervals with the Selberg-Delange method. As an application, we generalize the Deshouillers-Dress-Tenenbaum arcsine law on divisors to the short interval case.
The semigroup of nonempty finite subsets of integers.
The semigroup of nonempty finite subsets of rationals.
The sequence of fractional parts of roots
We study the function , where θ is a positive real number, ⌊·⌋ and · are the floor and fractional part functions, respectively. Nathanson proved, among other properties of , that if log θ is rational, then for all but finitely many positive integers n, . We extend this by showing that, without any condition on θ, all but a zero-density set of integers n satisfy . Using a metric result of Schmidt, we show that almost all θ have asymptotically (log θ log x)/12 exceptional n ≤ x. Using continued...
The sequences of prime divisors of integers
The set of minimal distances in Krull monoids
Let H be a Krull monoid with class group G. Then every nonunit a ∈ H can be written as a finite product of atoms, say . The set (a) of all possible factorization lengths k is called the set of lengths of a. If G is finite, then there is a constant M ∈ ℕ such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference d ∈ Δ*(H), where Δ*(H) denotes the set of minimal distances of H. We show that max Δ*(H) ≤ maxexp(G)-2,(G)-1 and that equality holds if every...
The set of rational cycles for the 3x+1 problem
The set of solutions of a polynomial-exponential equation