The greatest common divisor of certain sets of binomial coefficients

A formula is obtained for the greatest common divisor of any number of consecutive terms in any given row of Pascal's triangle.     

Rings with the greatest common divisor

We have found some class between the class of associative commutative factorial rings and the class of domains of integrity. It is the class of GCD-rings or almost factorial rings, i.e., domains of integrity where any two elements have a greatest common divisor.     

最大公因数矩阵的行列式

设Ｓ＝｛ｘ１，ｘ２，…，ｘｎ）是含ｎ个不同正整数的集合，（Ｓ）表示定义在Ｓ上的最大公因数矩阵，本文证明了且等号成立当且仅当Ｓ是最大公因数封闭集．     

Multidimensional greatest common divisor and Lehmer algorithms

A class of multidimensional greatest common divisor algorithms is studied. Their connection with the Jacobi algorithm is established and used to obtain theoretical properties such as the existence of digit frequencies. A technique of D. H. Lehmer's for Euclid's algorithm is generalized for efficient computation of the multidimensional algorithms. For triples of integers, two algorithms of interest are studied empirically.This work was partially supported by NSF grant #DCR75-07070.     

On the distribution of values of certain divisor functions

Let {?_d} be a sequence of nonnegative numbers and f(n) = Σ?_d, the sum being over divisors d of n. We say that f has the distribution function F if for all c ≥ 0, the number of integers n ≤ x for which f(n) > c is asymptotic to xF(c), and we investigate when F exists and when it is continuous.     

On the local distribution of certain arithmetic functions

Let d(n), σ ₁(n), and φ(n) stand for the number of positive divisors of n, the sum of the positive divisors of n, and Euler’s function, respectively. For each ν ∈, Z, we obtain asymptotic formulas for the number of integers n ⩽ x for which e _n = 2 ^v r for some odd integer m as well as for the number of integers n ⩽ x for which e _n = 2 ^v r for some odd rational number r. Our method also applies when φ(n) is replaced by σ ₁(n), thus, improving upon an earlier result of Bateman, Erdős, Pomerance, and Straus, according to which the set of integers n such that is an integer is of density 1/2. Research supported in part by a grant from NSERC. Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 3, pp. 315–331, July–September, 2006.     

On computation of the greatest common divisor of several polynomials over a finite field

We propose a probabilistic algorithm to reduce computing the greatest common divisor of m polynomials over a finite field (which requires computing m−1 pairwise greatest common divisors) to computing the greatest common divisor of two polynomials over the same field.     

General divisor functions in arithmetic progressions to large moduli

We prove a result on the distribution of the general divisor functions in arithmetic progressions to smooth moduli which exceed the square root of the length.     

Symbolic calculation of greatest common divisor of 2D polynomial matrices

An automatic process by which the greatest common divisor of2D polynomial matrices can be calculated using the symboliccomputation package MAPLE.     

On locally repeated values of certain arithmetic functions,I

Let ν(n) denote the number of distinct prime factors of n. We show that the equation n + ν(n) = m + ν(m) has many solutions with n ≠ m. We also show that if ν is replaced by an arbitrary, integer-valued function f with certain properties assumed about its average order, then the equation n + f(n) = m + f(m) has infinitely many solutions with n ≠ m.     

Asymptotic formulas for certain arithmetic functions

This is an extended summary of a talk given by the last named author at the Czecho-Slovake Number Theory Conference 2005, held at Malenovice in September 2005. It surveys some recent results concerning asymptotics for a class of arithmetic functions, including, e.g., the second moments of the number-of-divisors function d(n) and of the function r(n) which counts the number of ways to write a positive integer as a sum of two squares. For the proofs, reference is made to original articles by the authors published elsewhere. The last named author gratefully acknowledges support from the Austrian Science Fund (FWF) under project Nr. P18079-N12.     

On the oscillatory behavior of certain arithmetic functions associated with automorphic forms

We establish the oscillatory behavior of several significant classes of arithmetic functions that arise (at least presumably) in the study of automorphic forms. Specifically, we examine general L-functions conjectured to satisfy the Grand Riemann Hypothesis, Dirichlet series associated with classical entire forms of real weight and multiplier system, Rankin-Selberg convolutions (both “naive” and “modified”), and spinor zeta-functions of Hecke eigenforms on the Siegel modular group of genus two. For the second class we extend results obtained previously and jointly by M. Knopp, W. Kohnen, and the author, whereas for the fourth class we provide a new proof of a relatively recent result of W. Kohnen.