scholarly journals Factorization theorems for relatively prime divisor sums, GCD sums and generalized Ramanujan sums

2020 ◽  
Author(s):  
Hamed Mousavi ◽  
Maxie D. Schmidt
Keyword(s):  
Prime Divisor ◽  
Ramanujan Sums ◽  
Divisor Sums

On some punctured codes of ℤq-Simplex codes

2021 ◽  
pp. 2250012
Author(s):  
J. Prabu ◽  
J. Mahalakshmi ◽  
C. Durairajan ◽  
S. Santhakumar
Keyword(s):  
Prime Divisor ◽  
Weight Distribution ◽  
Linear Code ◽  
Prime Power ◽  
Punctured Codes ◽  
New Codes ◽  
Simplex Codes ◽  
Simplex Code ◽  
Partial Weight ◽  
Unit Formula

In this paper, we have constructed some new codes from [Formula: see text]-Simplex code called unit [Formula: see text]-Simplex code. In particular, we find the parameters of these codes and have proved that it is a [Formula: see text] [Formula: see text]-linear code, where [Formula: see text] and [Formula: see text] is a smallest prime divisor of [Formula: see text]. When rank [Formula: see text] and [Formula: see text] is a prime power, we have given the weight distribution of unit [Formula: see text]-Simplex code. For the rank [Formula: see text] we obtain the partial weight distribution of unit [Formula: see text]-Simplex code when [Formula: see text] is a prime power. Further, we derive the weight distribution of unit [Formula: see text]-Simplex code for the rank [Formula: see text] [Formula: see text].

scholarly journals Ergodic averages with prime divisor weights in

2017 ◽  
Vol 39 (4) ◽  
pp. 889-897 ◽  
Author(s):  
ZOLTÁN BUCZOLICH
Keyword(s):  
Dynamical System ◽  
Ergodic Theorem ◽  
Prime Divisor ◽  
Ergodic Averages ◽  
Weighting Functions ◽  
Pointwise Ergodic Theorem ◽  
Prime Factors ◽  
Ergodic Dynamical System ◽  
Image Position

We show that $\unicode[STIX]{x1D714}(n)$ and $\unicode[STIX]{x1D6FA}(n)$, the number of distinct prime factors of $n$ and the number of distinct prime factors of $n$ counted according to multiplicity, are good weighting functions for the pointwise ergodic theorem in $L^{1}$. That is, if $g$ denotes one of these functions and $S_{g,K}=\sum _{n\leq K}g(n)$, then for every ergodic dynamical system $(X,{\mathcal{A}},\unicode[STIX]{x1D707},\unicode[STIX]{x1D70F})$ and every $f\in L^{1}(X)$, $$\begin{eqnarray}\lim _{K\rightarrow \infty }\frac{1}{S_{g,K}}\mathop{\sum }_{n=1}^{K}g(n)f(\unicode[STIX]{x1D70F}^{n}x)=\int _{X}f\,d\unicode[STIX]{x1D707}\quad \text{for }\unicode[STIX]{x1D707}\text{ almost every }x\in X.\end{eqnarray}$$ This answers a question raised by Cuny and Weber, who showed this result for $L^{p}$, $p>1$.

scholarly journals On a density problem of Erdös

1999 ◽  
Vol 22 (3) ◽  
pp. 655-658 ◽  
Author(s):  
Safwan Akbik
Keyword(s):  
Positive Integer ◽  
Prime Divisor ◽  
Density Problem

For a positive integern, letP(n)denotes the largest prime divisor ofnand define the set:𝒮(x)=𝒮={n≤x:n   does not divide   P(n)!}. Paul Erdös has proposed that|S|=o(x)asx→∞, where|S|is the number ofn∈S. This was proved by Ilias Kastanas. In this paper we will show the stronger result that|S|=O(xe−1/4logx).

Weighted divisor sums and Bessel function series, IV

2012 ◽  
Vol 29 (1-3) ◽  
pp. 79-102 ◽  
Author(s):  
Bruce C. Berndt ◽  
Sun Kim ◽  
Alexandru Zaharescu
Keyword(s):  
Bessel Function ◽  
Function Series ◽  
Divisor Sums

scholarly journals Corrigenda On a classification of the function fields of algebraic tori: (Nagoya Math. J. 56 (1975), 85–104)

1980 ◽  
Vol 79 ◽  
pp. 187-190 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Prime Divisor ◽  
Dihedral Group ◽  
Quaternion Group ◽  
Algebraic Tori ◽  
Root Of Unity ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.

scholarly journals A geometric approach to Quillen’s conjecture

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Antonio Díaz Ramos ◽  
Nadia Mazza
Keyword(s):  
Finite Group ◽  
Prime Divisor ◽  
Geometric Approach ◽  
Classical Groups ◽  
Strong Version ◽  
Finite Classical Groups ◽  
A Finite Group

Abstract We introduce admissible collections for a finite group 𝐺 and use them to prove that most of the finite classical groups in non-defining characteristic satisfy the Quillen dimension at 𝑝 property, a strong version of Quillen’s conjecture, at a given odd prime divisor 𝑝 of | G | \lvert G\rvert . Compared to the methods in [M. Aschbacher and S. D. Smith, On Quillen’s conjecture for the 𝑝-groups complex, Ann. of Math. (2) 137 (1993), 3, 473–529], our techniques are simpler.

THE KORSELT SET OF pq

2012 ◽  
Vol 08 (02) ◽  
pp. 299-309 ◽  
Author(s):  
OTHMAN ECHI ◽  
NEJIB GHANMI
Keyword(s):  
Positive Integer ◽  
Prime Divisor ◽  
Prime Numbers ◽  
Composite Number

Let α ∈ ℤ\{0}. A positive integer N is said to be an α-Korselt number (Kα-number, for short) if N ≠ α and N - α is a multiple of p - α for each prime divisor p of N. By the Korselt set of N, we mean the set of all α ∈ ℤ\{0} such that N is a Kα-number; this set will be denoted by [Formula: see text]. Given a squarefree composite number, it is not easy to provide its Korselt set and Korselt weight both theoretically and computationally. The simplest kind of squarefree composite number is the product of two distinct prime numbers. Even for this kind of numbers, the Korselt set is far from being characterized. Let p, q be two distinct prime numbers. This paper sheds some light on [Formula: see text].

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