Footnote to a Formula of Gioia and Subbarao

1967 ◽  
Vol 10 (5) ◽  
pp. 749-750
Author(s):  
S. L. Segal
Keyword(s):  
Explicit Formula ◽  
Arithmetic Function ◽  
Fixed Integer ◽  
Simple Idea ◽  
Image Position ◽  
Multiplicative Formula

Recently Gioia and Subbarao [2] studied essentially the following problem: If g(n) is an arithmetic function, and , then what is the behaviour of H(a, n) defined for each fixed integer a ≥ 2 by1By using Vaidyanathaswamy′s formula [e.g., 1], they obtain an explicit formula for H(a, n) in case g(n) is positive and completely multiplicative (Formula 2.2 of [2]). However, Vaidyanathaswamy′s formula is unnecessary to the proof of this result, which indeed follows more simply without its use, by exploiting a simple idea used earlier by Subbarao [3] (referred to also in the course of [2]).

On General Polynomials

1967 ◽  
Vol 10 (4) ◽  
pp. 579-583 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Finite Field ◽  
Number Of Solutions ◽  
Fixed Integer ◽  
Image Position

Let d denote a fixed integer > 1 and let GF(q) denote the finite field of q = pn elements. We consider q fixed ≥ A(d), where A(d) is a (large) constant depending only on d. Let1where each aiεGF(q). Let nr(r = 2, 3, …, d) denote the number of solutions in GF(q) offor which x1, x2, …, xr are all different.

Generating Functions for a Class of Arithmetic Functions

1966 ◽  
Vol 9 (4) ◽  
pp. 427-431 ◽  
Author(s):  
A. A. Gioia ◽  
M.V. Subbarao
Keyword(s):  
Generating Functions ◽  
Multiplicative Function ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Prime Divisors ◽  
Image Position

In this note the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n ≥ 1, and L(l) = 0, w(l) = 1. An arithmetic function f is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1. It is known ([1], [3], [4]) that every multiplicative function f satisfies the identity1.1

scholarly journals Distribution of unitarily k-free integers

1975 ◽  
Vol 20 (2) ◽  
pp. 129-141 ◽  
Author(s):  
D. Suryanarayana ◽  
R. Sita Rama Chandra Rao
Keyword(s):  
Error Term ◽  
Real Variable ◽  
Canonical Representation ◽  
Order Estimate ◽  
Fixed Integer ◽  
Unitary Divisor ◽  
The Riemann Zeta Function ◽  
Elementary Methods ◽  
Riemann Zeta ◽  
Image Position

Let k be a fixed integer ≧2. A positive integer n is called unitarily k-free, if the multiplicity of each prime divisor of n is not a multiple of k; or equivalently, if n is not divisible unitarily by the kth power of any integer > 1. By a unitary divisor, we mean as usual a divisor d > 0 of n such that (d,(n/d)) = 1. The integer 1 is also considered to be unitarily k-free. These integers were first defined by Cohen (1961; § 1). Let Q*k denote the set of unitarily k-free integers. When k = 2, the set Q*2 coincides with the set Q* of exponentially odd integers (that is, integers in whose canonical representation each exponent is odd) discussed by Cohen himself in an earlier paper (1961; § 1 and § 6). Let x denote a real variable 1 and let Q*k denote the number of unitarily k-free integers ≦ x. Cohen (1961; Theorem 3.2) established by purely elementary methods that , where , the product being extended over all primes p and ζ(k) denotes the Riemann Zeta function. In the same paper Cohen (1961; Theorem 4.2) improved the order estimate of the error term in (1.1) to O(x1/k), by making use of the properties of real Dirichiet series. Later, he (Cohen; 1964) proved the same result by purely elementary methods eliminating the use of Dirichlet series.

On the decimal representation of integers

1952 ◽  
Vol 48 (4) ◽  
pp. 555-565 ◽  
Author(s):  
M. P. Drazin ◽  
J. Stanley Griffith
Keyword(s):  
Positive Integer ◽  
Fixed Integer ◽  
Usual Convention ◽  
Representation Of Integers ◽  
Image Position

Let r be any fixed integer with, r≥ 2; then, given any positive integer n, we can find* integers αk(r, n) (k = 0, 1, 2, …) such thatwhere, subject to the conditionsthe integers αk(r, n) are uniquely determined, and, in fact, clearlyαk(r, n) = [n/rk] − r[n/rk+1](square brackets denoting integral parts, according to the usual convention).

scholarly journals Uniqueness criterion for a moment problem

1974 ◽  
Vol 18 (3) ◽  
pp. 303-305 ◽  
Author(s):  
A. Lenard
Keyword(s):  
Explicit Formula ◽  
Moment Problem ◽  
Uniqueness Criterion ◽  
The Moment ◽  
Image Position

In an article that appeared some years ago in this journal, Takács [1] gave a uniqueness criterion for the solution of the moment problem where the Br are given numbers and the Pk are sought, Pk ≧ 0, σkPk = 1. Takács showed that if Br < ∞ for all r and then the solution is unique. In addition, he gave an explicit formula for the solution in this case where q is any number satisfying q ≧ 0 and q > p2 — 1.

On average values of arithmetic functions

1941 ◽  
Vol 37 (4) ◽  
pp. 358-372 ◽  
Author(s):  
E. Fogels
Keyword(s):  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Average Order ◽  
Image Position

The problem considered in this paper is that of finding the least possible h = h(x) such that a given arithmetic function a(n) should keep its average order in the interval x, x + h, i.e. that we haveandas x → ∞.

Weak and strong law results for a function of the spacings

1985 ◽  
Vol 22 (03) ◽  
pp. 543-555
Author(s):  
William P. McCormick
Keyword(s):  
Order Statistics ◽  
Random Variables ◽  
Weak Limit ◽  
Strong Law ◽  
Fixed Integer ◽  
Image Position

Let be i.i.d. uniform on (0,1) random variables and define Si,n = Ui ,n–1 Ui– 1,n–1, i = 1, · ··, n where the Ui –n–1 are the order statistics from a sample of size n – 1 and U 0,n–1 =0 and Un,n– 1 = 1. The Si,n are called the spacings divided by U 1,· ··,Un– 1. For a fixed integer l, set . Exact and weak limit results are obtained for the Ml,n. Further we show that with probability 1 This extends results of Cheng.

scholarly journals On a subgroup of the group of multiplicative arithmetic functions

1975 ◽  
Vol 20 (3) ◽  
pp. 348-358 ◽  
Author(s):  
T. B. Carroll ◽  
A. A. Gioia
Keyword(s):  
Abelian Group ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Dirichlet Convolution ◽  
Image Position ◽  
Positive Integers ◽  
Multiplicative Arithmetic

An arithmetic function f is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n. Furthermore an arithmetic function is said to be linear (or completely multiplicative) if f(1) = 1 and f(mn) = f(m)f(n) for all positive integers m and n.The Dirichlet convolution of two arithmetic functions f and g is defined by for all n∈Z+. Recall that the set of all multiplicative functions, denoted by M, with this operation is an abelian group.

scholarly journals Further Results on the Construction of Mutually Orthogonal Latin Squares and the Falsity of Euler's Conjecture

1960 ◽  
Vol 12 ◽  
pp. 189-203 ◽  
Author(s):  
R. C. Bose ◽  
S. S. Shrikhande ◽  
E. T. Parker
Keyword(s):  
Prime Power ◽  
Arithmetic Function ◽  
Latin Squares ◽  
Orthogonal Latin Squares ◽  
Mutually Orthogonal Latin Squares ◽  
Image Position

Ifis the prime power decomposition of an integer v, and we define the arithmetic function n(v) bythen it is known, MacNeish (10) and Mann (11), that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. We shall denote by N(v) the maximum possible number of mutually orthogonal Latin squares of order v. Then the Mann-MacNeish theorem can be stated asMacNeish conjectured that the actual value of N(v) is n(v).

A non-nilpotent Lie ring satisfying the Engel condition and a non-nilpotent Engel group

1955 ◽  
Vol 51 (3) ◽  
pp. 401-405 ◽  
Author(s):  
P. M. Cohn
Keyword(s):  
Fixed Integer ◽  
Lie Ring ◽  
Engel Group ◽  
Image Position ◽  
Engel Condition

Let L be a Lie ring and denote the product of x and y in L by [x, y]. The ring L is said to satisfy the Engel condition (cf. (1)), if for every pair of elements x, yεL there is an integer k = k(x, y)such thatIf k(x, y) can be taken equal to a fixed integer n for all x, y ε L then L is said to satisfy the n-th Engel condition.

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