Showing papers on "Divisor published in 1970"

Dirichlet’s divisor problem

Abstract: Let d(n) denote the number of positive divisors of the positive integer n. Let $$E(x) = \sum\limits_{n \leqslant x} {d(n) - x\log x - (2\gamma - 1)x,\,x \geqslant 1}$$ where γ is Euler’s constant. It is known, after Dirichlet, that $$ E(x) = 0({x^{{\frac{1}{2}}}}),\quad as\quad x \to \infty $$

On canonically polarized varieties (ii).

Abstract: prime numbers and 1 , M; (P2) The set (pi, ,p.) consists of the prime divisors of the order of the torsion group of V-divisors and the characteristic of the universal domain; (P3) $ contains a non-degenerate divisor 2 X; (P4) $ consists of those divisors Y such that rX and sY are algebraically equivalent on V with r, s , M.3'4 Then we called (V, %) a polarized variety (of type M) and proved the existence of a basic polar divisor Vv in [3]. Let ,t (tXv) be the invertible sheaf defined by tXv. We also called the polynomial P(t) = x(V,1e(tXv)) in t the Hilbert characteristic polynomial of (V, $) and showed that an algebraic deformation (V1, W') of (V, $) has the same Hiilbert characteristic polynomial as (V, $) as long as both have the same ranic (i.e. Xv(n)Xv(n)

A multiple-precision division algorithm

Abstract: A generalized division algorithm for use with positive integral operands is presented. Depending upon the algebraic relationship of the first two ciphers of the divisor, one or at most two adjustments to the original divisor and dividend must be performed before the division operation can be initiated. The uniqueness of this method will cause each trial cipher in the quotient to be either equal to or one greater than its final replacement.

On the Penultimate Remainder Algorithm and the Catalytic Multiplier

Abstract: Given monic polynomials, f(z) and p(z), of degrees n and p < n, respectively, iff(z) is divided by p(z) up to but not including the constant term in the division, and if suitable conditions are satisfied, then the remainder, which is also of degree p, will more closely approximate a true divisor off(z) than does p(z). This is Lin's penultimate remainder algorithm [3]. Unfortunately, when the conditions are not satisfied the remainder can deviate even further from a true divisor, even though p(z) is close to one. Aitken [1], [2] showed that by proper choice of h(z), if the division is made into h(z)f(z), instead of into f(z), then the penultimate remainder will always closely approximate a true divisor, provided only p(z) is itself sufficiently close. This h(z) is Aitken's catalytic multiplier. Lin's development is complicated and unsuggestive. Aitken's derivation of the conditions for convergence is very sketchy and leaves much to the reader. In particular, it depends upon a lemma not explicitly stated and apparently not generally known. It is the purpose of this note to state and prove this lemma, and then to show how these conditions come out quite naturally from it. LEMMA. Let