Orthonormal series and density of integers

Selectivity Estimation of Range Queries Based on Data Density Approximation via Orthonormal Series

Selectivity estimation is an integral part of query optimization. In this paper, we propose to approximate data density functions of relations and use the approximations to estimate selectivities of range queries. A data density function here is approximated by a partial sum of an orthonormal series. Compared with histogram-based approaches, such as the wavelet, DCT, and kernel-spline methods, ...

Random Series in Powers of Algebraic Integers

Density of Singular Pairs of Integers

A positive integer n is called cyclic if there is a unique group of order n, which is necessarily cyclic. Using a characterisation of the cyclic integers as those n satisfying gcd(n, φ(n)) = 1, P. Erdős (1947) proved that the number of cyclic integers n ≤ x is asymptotic to z(x) = e−γ x log log log x , as x→∞, where γ is Euler’s constant. An ordered pair of integers (m,n) is called singular if ...

Bounds for the Density of Abundant Integers

Introduction 1. Expressing A(x) as a Sum 2. Trivial Bounds for Ak(x) 3. Lower Bound for A(x) 4. Upper Bounds for Ak(x) 5. Mean Values of f(n) and Upper Bounds for Ak(x) 6. Upper Bounds for the Euler Products V k(r) 7. Numerical Results 8. Other Experimental Results References We say that an integer n is abundant if the sum of the divisors of n is at least 2n. It has been known [Wall 1972] that ...

On a Density for Sets of Integers

A relationship between the Riemann zeta function and a density on integer sets is explored. Several properties of the introduced density are derived.