In 1849, Dirichlet [5] proved that the probability that two positive integers are relatively prime is 1/ζ(2). Later, it was generalized into the case that positive integers has no nontrivial kth power common divisor. In this paper, we further generalize this result: the probability that the gcd of m products of n positive integers is B-friable is ∏ p>B [ 1− { 1− ( 1− 1 p )n}m] for m ≥ 2. We sho...

Let ∗ be a star operation on an integral domain D. Let f(D) be the set of all nonzero finitely generated fractional ideals of D. Call D a ∗–Prüfer (respectively, (∗, v)–Prüfer) domain if (FF−1)∗ = D (respectively, (F vF−1)∗ = D) for all F ∈ f(D). We establish that ∗–Prüfer domains (and (∗, v)–Prüfer domains) for various star operations ∗ span a major portion of the known generalizations of Prüf...

We propose a fast algorithm for computing approximate GCD of univariate polynomials with coefficients that are given only to a finite accuracy. The algorithm is based on a stabilized version of the generalized Schur algorithm for Sylvester matrix and its embedding. All computations can be done in O(n2) operations, where n is the sum of the degrees of polynomials. The stability of the algorithm ...

In this paper, we develop a fast structured total least squares (STLS) algorithm for computing an approximate greatest common divisor (GCD) of two univariate polynomials. By exploiting the displacement structure of the Sylvester matrix and applying the generalized Schur algorithm, each single iteration of the proposed algorithm has quadratic computational complexity in the degrees of the given ...

The discrete Fourier transform of the greatest common divisor � id[a](m) = m � k=1 gcd(k,m)α m , with αm a primitive m-th root of unity, is a multiplicative function that generalizes both the gcd-sum function and Euler’s totient function. On the one hand it is the Dirichlet convolution of the identity with Ramanujan’s sum, � id[a] = id ∗ c•(a), and on the other hand it can be written as a gener...

We prove tight bounds for Sort, Merge, Insert, Gcd of integers, Gcd of polynomials and Rational functions with finite inputs domain in random access machine with arithmetic operations, direct and indirect addressing, unlimited power of answering YES/NO questions, branching and tables with bounded size. These bounds are also true even if we do not count additions, subtractions, multiplication an...

In this correspondence, a new viewpoint is proposed for estimating an image from its distorted versions in presence of noise without the a priori knowledge of the distortion functions. In z-domain, the desired image can be regarded as the greatest common polynomial divisor among the distorted versions. With the assumption that the distortion filters are finite impulse response (FIR) and relativ...