On the Sums Running over Reduced Residue Classes Evaluated at Polynomial Arguments

2020 ◽  
Author(s):  
Mehdi Hassani ◽  
Mahmoud Marie Marie
Keyword(s):  
Euler Function ◽  
Fixed Integer ◽  
Residue Classes

For a given polynomial G we study the sums φm(n) := ∑′km and φG(n) = ∑′G(k) where m ≥ 0 is a fixed integer and ∑′ runs through all integers k with 1 ≤ k ≤ n and gcd(k, n) = 1. Although, for m ≥ 1 the function φm is not multiplicative, analogue to the Euler function we obtain expressions for φm(n) and φG(n). Also, we estimate the averages ∑n≤x φm(n) and ∑n≤xφG(n), as more as, the alternative averages ∑n≤x(−1)n−1φm(n) and ∑n≤x(−1)n−1φG(n).

scholarly journals Limiting Properties of the Distribution of Primes in an Arbitrarily Large Number of Residue Classes

2020 ◽  
Vol 63 (4) ◽  
pp. 837-849 ◽  
Author(s):  
Lucile Devin
Keyword(s):  
Riemann Hypothesis ◽  
Linear Independence ◽  
Classical Case ◽  
Weak Version ◽  
Distribution Of Primes ◽  
Logarithmic Density ◽  
Fixed Integer ◽  
Residue Classes ◽  
Independence Hypothesis ◽  
And Function

AbstractWe generalize current known distribution results on Shanks–Rényi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function field analogues developed in the recent years. More precisely, let $\unicode[STIX]{x1D70B}(x;q,a)$ be the number of primes up to $x$ that are congruent to $a$ modulo $q$. For a fixed integer $q$ and distinct invertible congruence classes $a_{0},a_{1},\ldots ,a_{D}$, assuming the generalized Riemann Hypothesis and a weak version of the linear independence hypothesis, we show that the set of real $x$ for which the inequalities $\unicode[STIX]{x1D70B}(x;q,a_{0})>\unicode[STIX]{x1D70B}(x;q,a_{1})>\cdots >\unicode[STIX]{x1D70B}(x;q,a_{D})$ are simultaneously satisfied admits a logarithmic density.

scholarly journals Théorèmes de type Fouvry–Iwaniec pour les entiers friables

2015 ◽  
Vol 151 (5) ◽  
pp. 828-862 ◽  
Author(s):  
Sary Drappeau
Keyword(s):  
Recent Work ◽  
Prime Factor ◽  
Arithmetic Progressions ◽  
Dispersion Method ◽  
Fixed Integer ◽  
Residue Classes

An integer $n$ is said to be $y$-friable if its largest prime factor $P^{+}(n)$ is less than $y$. In this paper, it is shown that the $y$-friable integers less than $x$ have a weak exponent of distribution at least $3/5-{\it\varepsilon}$ when $(\log x)^{c}\leqslant x\leqslant x^{1/c}$ for some $c=c({\it\varepsilon})\geqslant 1$, that is to say, they are well distributed in the residue classes of a fixed integer $a$, on average over moduli ${\leqslant}x^{3/5-{\it\varepsilon}}$ for each fixed $a\neq 0$ and ${\it\varepsilon}>0$. We apply this to the estimation of the sum $\sum _{2\leqslant n\leqslant x,P^{+}(n)\leqslant y}{\it\tau}(n-1)$ when $(\log x)^{c}\leqslant y$. This follows and improves on previous work of Fouvry and Tenenbaum. Our proof combines the dispersion method of Linnik in the setting of Bombieri, Fouvry, Friedlander and Iwaniec with recent work of Harper on friable integers in arithmetic progressions.

A Transferable, Polarisable Force Field for Ionic Liquids

2019 ◽  
Author(s):  
Kateryna Goloviznina ◽  
José N. Canongia Lopes ◽  
Margarida Costa Gomes ◽  
Agilio Padua
Keyword(s):  
Ionic Liquids ◽  
Force Field ◽  
Force Fields ◽  
Dipole Moments ◽  
Van Der Waals Interactions ◽  
First Principles Calculations ◽  
Ion Diffusion ◽  
Fixed Integer ◽  
Molecular Compounds ◽  
Induced Dipoles

A general, transferable polarisable force field for molecular simulation of ionic liquids and their mixtures with molecular compounds is developed. This polarisable model is derived from the widely used CL\&P fixed-charge force field that describes most families of ionic liquids, in a form compatible with OPLS-AA, one of the major force fields for organic compounds. Models for ionic liquids with fixed, integer ionic charges lead to pathologically slow dynamics, a problem that is corrected when polarisation effects are included explicitly. In the model proposed here, Drude induced dipoles are used with parameters determined from atomic polarisabilities. The CL\&P force field is modified upon inclusion of the Drude dipoles, to avoid double-counting of polarisation effects. This modification is based on first-principles calculations of the dispersion and induction contributions to the van der Waals interactions, using symmetry-adapted perturbation theory (SAPT) for a set of dimers composed of positive, negative and neutral fragments representative of a wide variety of ionic liquids. The fragment approach provides transferability, allowing the representation of a multitude of cation and anion families, including different functional groups, without need to re-parametrise. Because SAPT calculations are expensive an alternative predictive scheme was devised, requiring only molecular properties with a clear physical meaning, namely dipole moments and atomic polarisabilities. The new polarisable force field, CL\&Pol, describes a broad set set of ionic liquids and their mixtures with molecular compounds, and is validated by comparisons with experimental data on density, ion diffusion coefficients and viscosity. The approaches proposed here can also be applied to the conversion of other fixed-charged force fields into polarisable versions.<br>

scholarly journals N-th root rings

1987 ◽  
Vol 35 (1) ◽  
pp. 111-123 ◽  
Author(s):  
Henry Heatherly ◽  
Altha Blanchet
Keyword(s):  
Commutative Ring ◽  
Subdirect Product ◽  
General Construction ◽  
Fixed Integer ◽  
Chain Conditions

A ring for which there is a fixed integer n ≥ 2 such that every element in the ring has an n-th in the ring is called an n-th root ring. This paper gives numerous examples of diverse types of n-th root rings, some via general construction procedures. It is shown that every commutative ring can be embedded in a commutative n-th root ring with unity. The structure of n-th root rings with chain conditions is developed and finite n-th root rings are completely classified. Subdirect product representations are given for several classes of n-th root rings.

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