Abstract: A modulo arithmetic unit and method for providing a sum of first and second numbers is provided. In one form, a first adder calculates a first sum which is equal to the arithmetic sum of the first and second numbers. A second adder is provided for adding the first number to an offset value equal to (2 X -M), where X defines the number of bits of the number system used, M is a predetermined modulus and X and M are integers. A third adder operates in parallel with the first adder to calculate the sum of the output value of the second adder and the second operand to provide a second output sum and a carry output bit. In another form, only two adders are utilized wherein the first adder calculates a first output sum of the first and second numbers, and the second adder calculates the sum of the first output sum and the offset value. Both illustrated forms utilize a multiplexer which outputs one of the two calculated output sums depending upon whether a wraparound of an upper modulus boundary occurred.

Abstract: Let dk(n) be the number of the factorizations of n into k positive numbers. It is known that the following asymptotic formula holds: where r and q are co-prime integers with 0 < r < q, Pk is a polynomial of degree k − 1, φ(q) is the Euler function, and Δk (q; r) is the error term. (See Lavrik [3]).

Abstract: This paper is concerned with estimating the number of positive integers up to some bound (which tends to infinity), such that they have a fixed number of prime divisors, and lie in a given arithmetic progression. We obtain estimates which are uniform in the number of prime divisors, and at the same time, in the modulus of the arithmetic progression. These estimates take the form of a fixed but arbitrary number of main terms, followed by an error term.

Abstract: so. Introduction In this paper we obtain upper bounds, and in some cases even asymptotic estimates, for the moments of additive functions f(n), where n belongs to a subset S of the positive integers ;Z+ satisfying some properties to be specified later. We also discuss certain consequences of these estimates. The theorems stated in §2 extend various classical results to such subsets; nevertheless the main point here is the method we employ, which is new. Our method stems from a recent technique due to Elliott [6] who obtained uniform upper bounds for the moments of arbitrary additive functions in the case S = ;Z+. We will show that by employing the combinatorial sieve and the bilateral Laplace transform along with some of Elliott's ideas one obtains a substantially improved method. For a class of sets S we can derive similar upper bounds for the absolute moments of all complex valued additive functions. In addition we can evaluate the moments asymptotically provided the f(n) satisfy some conditions. From these asymptotic estimates we get information concerning the distribution function of such f(n), for n € S. There is a vast literature on the distribution of additive functions and a variety of methods available (see Elliott [5], Vols. I and 11). One approach, which is due to Halber­ stam[l3: I, II, Ill], makes use of the method of moments to determine the limiting distribution of certain additive functions. Despite the difficult calculations it involved, this method had the attraction of being elementary and so capable of wide application. Subsequently this method underwent simplification and refinement by Delange [3], [4]. Being based upon the combinatorial sieve, which is an elementary tool, our method retains the applicability of Halberstam's approach, but without the latter's complications, since our use of the bilateral Laplace transform introduces the necessary simplification.

Abstract: Ew obtain an asymptotic formula and a theorem about the mean (of the type of the “large sieve”) for the numberF c,d (x;q,l) of primes p⩽x such thatp=l(modq), p=[tc]=[n d ], t,n e ℤ, whereq>0, l∈ℤ,c,d ∈ ℝ are given numbers.

Abstract: A product ofk≥k 0 (d) consecutive members of an arithmetic progression of differenced cannot be a proper power.