scholarly journals Proof of a conjecture of Heath-Brown concerning quadratic residues

1996 ◽  
Vol 39 (3) ◽  
pp. 581-588 ◽  
Author(s):  
R. R. Hall
Keyword(s):  
General Result ◽  
Multiplicative Functions ◽  
Quadratic Residues ◽  
Bounded Below ◽  
Positive Integers

The conjecture in question is that the proportion of the first n positive integers which are quadratic residues of an arbitrary prime p is bounded below by a positive. δ. This is established here as a corollary of a more general result concerning multiplicative functions; the problem of the sharp δ is left open.

MORE ON A CERTAIN ARITHMETICAL DETERMINANT

2017 ◽  
Vol 97 (1) ◽  
pp. 15-25 ◽  
Author(s):  
ZONGBING LIN ◽  
SIAO HONG
Keyword(s):  
Linear Algebra ◽  
Acta Math ◽  
Arithmetical Function ◽  
Greatest Common Divisor ◽  
Multiplicative Functions ◽  
Common Multiple ◽  
Least Common Multiple ◽  
Mobius Function ◽  
Dirichlet Convolution ◽  
Positive Integers

Let $n\geq 1$ be an integer and $f$ be an arithmetical function. Let $S=\{x_{1},\ldots ,x_{n}\}$ be a set of $n$ distinct positive integers with the property that $d\in S$ if $x\in S$ and $d|x$. Then $\min (S)=1$. Let $(f(S))=(f(\gcd (x_{i},x_{j})))$ and $(f[S])=(f(\text{lcm}(x_{i},x_{j})))$ denote the $n\times n$ matrices whose $(i,j)$-entries are $f$ evaluated at the greatest common divisor of $x_{i}$ and $x_{j}$ and the least common multiple of $x_{i}$ and $x_{j}$, respectively. In 1875, Smith [‘On the value of a certain arithmetical determinant’, Proc. Lond. Math. Soc. 7 (1875–76), 208–212] showed that $\det (f(S))=\prod _{l=1}^{n}(f\ast \unicode[STIX]{x1D707})(x_{l})$, where $f\ast \unicode[STIX]{x1D707}$ is the Dirichlet convolution of $f$ and the Möbius function $\unicode[STIX]{x1D707}$. Bourque and Ligh [‘Matrices associated with classes of multiplicative functions’, Linear Algebra Appl. 216 (1995), 267–275] computed the determinant $\det (f[S])$ if $f$ is multiplicative and, Hong, Hu and Lin [‘On a certain arithmetical determinant’, Acta Math. Hungar. 150 (2016), 372–382] gave formulae for the determinants $\det (f(S\setminus \{1\}))$ and $\det (f[S\setminus \{1\}])$. In this paper, we evaluate the determinant $\det (f(S\setminus \{x_{t}\}))$ for any integer $t$ with $1\leq t\leq n$ and also the determinant $\det (f[S\setminus \{x_{t}\}])$ if $f$ is multiplicative.

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 On Integers n Relatively Prime To ƒ(n)

1955 ◽  
Vol 7 ◽  
pp. 155-158 ◽  
Author(s):  
Joachim Lambek ◽  
Leo Moser
Keyword(s):  
General Result ◽  
Positive Integers ◽  
Prime 2

1. Introduction. If m and n are two integers chosen at random, the probability that they are relatively prime (2, p. 267) is 6π-2. This result may still hold when m and n are functionally related. Thus, Watson (3) recently proved that for α irrational, the positive integers n for which (n, [αn]) = 1, have density 6π-2. A different proof of a slightly more general result was given by Estermann (1).

scholarly journals Powers in products of terms of Pell's and Pell–Lucas Sequences

2015 ◽  
Vol 11 (04) ◽  
pp. 1259-1274 ◽  
Author(s):  
Jhon J. Bravo ◽  
Pranabesh Das ◽  
Sergio Guzmán ◽  
Shanta Laishram
Keyword(s):  
General Result ◽  
Diophantine Equation ◽  
Initial Condition ◽  
Lucas Sequences ◽  
Lucas Sequence ◽  
Positive Integers

In this paper, we consider the usual Pell and Pell–Lucas sequences. The Pell sequence [Formula: see text] is given by the recurrence un = 2un-1 + un-2 with initial condition u0 = 0, u1 = 1 and its associated Pell–Lucas sequence [Formula: see text] is given by the recurrence vn = 2vn-1 + vn-2 with initial condition v0 = 2, v1 = 2. Let n, d, k, y, m be positive integers with m ≥ 2, y ≥ 2 and gcd (n, d) = 1. We prove that the only solutions of the Diophantine equation unun+d⋯un+(k-1)d = ym are given by u7 = 132 and u1u7 = 132 and the equation vnvn+d⋯vn+(k-1)d = ym has no solution. In fact, we prove a more general result.

scholarly journals A New Postulate and Some Conjectures Concerning Pair Primes in the Interval [n!, (n + k)!]

2020 ◽  
Vol 3 (1) ◽  
pp. 1-6
Author(s):  
Abiodun E. Adeyemi
Keyword(s):  
Prime Numbers ◽  
Future Research ◽  
Structural Form ◽  
Numerical Evidence ◽  
Bounded Below ◽  
Positive Integers

This paper rather studies the behaviour of prime numbers bounded below and above by positive integers n! and (n + k)!, and then after some numerical evidence, postulates that there is at least one pair primes of gap k ∈ 2Z+ in between n! and (n + k)! for every integer n ≥ 2 and every even integer k > 0. This assertion would eventually provide another structural form for Euclid theorem of ifinitude of primes, a kind of projection of the form in the original Bertrand postulate (now Chebychev's theorem). The truth- fulness of the conjecture that emanated from this postulate implies the Polignac's conjecture which aptly generalizes the twin prime conjecture. We thus present the new postulate and the conjectures for future research.

scholarly journals Multiplicative functions commutable with sums of squares

2018 ◽  
Vol 14 (02) ◽  
pp. 469-478 ◽  
Author(s):  
Poo-Sung Park
Keyword(s):  
Multiplicative Function ◽  
Sums Of Squares ◽  
Multiplicative Functions ◽  
Identity Function ◽  
Positive Integers

Let [Formula: see text] be an integer greater than or equal to [Formula: see text]. We show that if a multiplicative function [Formula: see text] satisfies [Formula: see text] for all positive integers [Formula: see text], then [Formula: see text] is the identity function.

scholarly journals On an open problem of Simsek concerning the computation of a family of special numbers

2019 ◽  
Vol 13 (1) ◽  
pp. 61-72 ◽  
Author(s):  
Aimin Xu
Keyword(s):  
General Result ◽  
Open Problem ◽  
Stirling Numbers ◽  
Discrete Math ◽  
Euler Numbers ◽  
Negative Order ◽  
Positive Integers

Simsek [Y. Simsek, New families of special numbers for computing negative order Euler numbers and related numbers and polynomials, Appl. Anal. Discrete Math. 12(2018), 1-35.] conjectured that B(d,k)=(kd + x1kd-1 + x2kd-2 +...+ xd-1k)2k-d; where x1,x2,..., xd-1; d are positive integers, and proposed the following open problem: (I) How can we compute the coefficients x1, x2, ... , xd-1? (II) Is it possible to find the function fd(x) = ??,k=1 B(d,k)xk? By using the familiar Stirling numbers of the first and second kind, we solve this problem. We further obtain a general result on the generalized numbers of B(d,k).

Some formulae for the G -function

1963 ◽  
Vol 59 (2) ◽  
pp. 347-350 ◽  
Author(s):  
R. K. Saxena
Keyword(s):  
General Result ◽  
Present Note ◽  
The Other ◽  
Positive Integers

The object of the present note is to evaluate some integrals involving Meijer's G-function, in which the argument of the G-function contains a factor where m and n are positive integers and t is the variable of integration. Two different forms of the general result have been obtained, one for m > n and the other for m < n. The value of the corresponding integral when m = n is also obtained. For the definition, properties and the behaviour of the G-function, see (2), §§ 5·3, 5·31 and (5), § 18.

Sharp Block–Sharkovsky Type Theorem for Multivalued Maps on the Circle and Its Application to Differential Equations and Inclusions

2019 ◽  
Vol 29 (09) ◽  
pp. 1950127 ◽  
Author(s):  
Jan Andres ◽  
Karel Pastor
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
General Result ◽  
Type Theorem ◽  
Multivalued Maps ◽  
Interval Maps ◽  
Translation Operators ◽  
Differential Equations And Inclusions ◽  
Positive Integers ◽  
Full Analogy

This is a final part of the series of our papers devoted to a multivalued version of the (Sharkovsky type) Block cycle coexistence theorem. It improves our last general result in the sense that its part related to the usual ordering of positive integers becomes a full analogy of the standard single-valued case, while the alternative part related to the Sharkovsky ordering of positive integers is an analogy of the multivalued case for interval maps, provided there exists a fixed point. That is why we call the obtained theorem here as “sharp”. This theorem is still applied via the associated Poincaré translation operators to differential equations and inclusions on the circle. All the deterministic results are also randomized in an advantageous way.

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