scholarly journals Resonance between the Representation Function and Exponential Functions over Arithemetic Progression

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Li Ma ◽  
Xiaofei Yan
Keyword(s):  
Positive Integer ◽  
Exponential Sum ◽  
Upper Bounds ◽  
Arithmetic Progressions ◽  
Large Parameter ◽  
Exponential Functions ◽  
Representation Function

Let r n denote the number of representations of a positive integer n as a sum of two squares, i.e., n = x 1 2 + x 2 2 , where x 1 and x 2 are integers. We study the behavior of the exponential sum twisted by r n over the arithmetic progressions ∑ n ∼ X n ≡ l mod q r n e α n β , where 0 ≠ α ∈ ℝ , 0 < β < 1 , e x = e 2 π i x , and n ∼ X means X < n ≤ 2 X . Here, X > 1 is a large parameter, 1 ≤ l ≤ q are integers, and l , q = 1 . We obtain the upper bounds in different situations.

Upper bounds on a two-term exponential sum*

2001 ◽  
Vol 44 (8) ◽  
pp. 1003-1015 ◽  
Author(s):  
Todd Cochrane ◽  
Zhiyong Zheng
Keyword(s):  
Exponential Sum ◽  
Upper Bounds

scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

On the Surjectivity of the Galois Representations Associated to Non-CM Elliptic Curves

2005 ◽  
Vol 48 (1) ◽  
pp. 16-31 ◽  
Author(s):  
Alina Carmen Cojocaru ◽  
Ernst Kani
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Elliptic Curves ◽  
Galois Representations ◽  
Complex Multiplication ◽  
Galois Representation ◽  
Upper Bounds

AbstractLet E be an elliptic curve defined over ℚ, of conductor N and without complex multiplication. For any positive integer l, let ϕl be the Galois representation associated to the l-division points of E. From a celebrated 1972 result of Serre we know that ϕl is surjective for any sufficiently large prime l. In this paper we find conditional and unconditional upper bounds in terms of N for the primes l for which ϕl is not surjective.

scholarly journals Integral quadratic forms avoiding arithmetic progressions

2020 ◽  
Vol 16 (10) ◽  
pp. 2141-2148
Author(s):  
A. G. Earnest ◽  
Ji Young Kim
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Integral Quadratic Form ◽  
Positive Integers

For every positive integer [Formula: see text], it is shown that there exists a positive definite diagonal quaternary integral quadratic form that represents all positive integers except for precisely those which lie in [Formula: see text] arithmetic progressions. For [Formula: see text], all forms with this property are determined.

scholarly journals Monochromatic arithmetic progressions with large differences

1999 ◽  
Vol 60 (1) ◽  
pp. 21-35
Author(s):  
Tom C. Brown ◽  
Bruce M. Landman
Keyword(s):  
Positive Integer ◽  
Lower Bounds ◽  
Arithmetic Progression ◽  
Constant Function ◽  
Arithmetic Progressions ◽  
Upper And Lower Bounds

A generalisation of the van der Waerden numbers w(k, r) is considered. For a function f: Z+ → R+ define w(f, k, r) to be the least positive integer (if it exists) such that for every r-coloring of [1, w(f, k, r)] there is a monochromatic arithmetic progression {a + id: 0 ≤ i ≤ k −1} such that d ≥ f(a). Upper and lower bounds are given for w(f, 3, 2). For k > 3 or r > 2, particular functions f are given such that w(f, k, r) does not exist. More results are obtained for the case in which f is a constant function.

REPRESENTATIONS OF ARITHMETIC PROGRESSIONS BY POSITIVE DEFINITE QUADRATIC FORMS

2011 ◽  
Vol 07 (06) ◽  
pp. 1603-1614 ◽  
Author(s):  
BYEONG-KWEON OH
Keyword(s):  
Positive Integer ◽  
Quadratic Form ◽  
Quadratic Forms ◽  
Positive Definite ◽  
Arithmetic Progressions ◽  
Negative Integer ◽  
Definite Integral ◽  
Integral Quadratic Form ◽  
Ternary Quadratic Forms

For a positive integer d and a non-negative integer a, let Sd,a be the set of all integers of the form dn + a for any non-negative integer n. A (positive definite integral) quadratic form f is said to be Sd,a-universal if it represents all integers in the set Sd, a, and is said to be Sd,a-regular if it represents all integers in the non-empty set Sd,a ∩ Q((f)), where Q(gen(f)) is the set of all integers that are represented by the genus of f. In this paper, we prove that there is a polynomial U(x,y) ∈ ℚ[x,y] (R(x,y) ∈ ℚ[x,y]) such that the discriminant df for any Sd,a-universal (Sd,a-regular) ternary quadratic forms is bounded by U(d,a) (respectively, R(d,a)).

ON MODAL μ-CALCULUS OVER FINITE GRAPHS WITH SMALL COMPONENTS OR SMALL TREE WIDTH

2012 ◽  
Vol 23 (03) ◽  
pp. 627-647
Author(s):  
GIOVANNA D'AGOSTINO ◽  
GIACOMO LENZI
Keyword(s):  
Positive Integer ◽  
Upper Bounds ◽  
Connected Components ◽  
Small Tree ◽  
Strongly Connected Components ◽  
Finite Graphs ◽  
Strongly Connected ◽  
Tree Width ◽  
Parity Games ◽  
Alternating Automata

This paper is a continuation and correction of a paper presented by the same authors at the conference GANDALF 2010. We consider the Modal μ-calculus and some fragments of it. For every positive integer k we consider the class SCCk of all finite graphs whose strongly connected components have size at most k, and the class TWk of all finite graphs of tree width at most k. As upper bounds, we show that for every k, the temporal logic CTL* collapses to alternation free μ-calculus in SCCk; and in TW1, the winning condition for parity games of any index n belongs to the level Δ2 of Modal μ-calculus. As lower bounds, we show that Büchi automata are not closed under complement in TW2 and coBüchi nondeterministic and alternating automata differ in TW1.

scholarly journals THE DISTRIBUTION OF r-FREE NUMBERS IN ARITHMETIC PROGRESSIONS

2014 ◽  
Vol 10 (03) ◽  
pp. 559-563 ◽  
Author(s):  
JASON GIBSON
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Arithmetic Progressions

Let r ≥ 2. A positive integer n is called r-free if n is not divisible by the r th power of a prime. Generalizing an earlier work of Orr, we provide an upper bound of Bombieri–Vinogradov type for the r-free numbers in arithmetic progressions.

scholarly journals Satisfiability and Computing van der Waerden Numbers

10.37236/1794 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Michael R. Dransfield ◽  
Lengning Liu ◽  
Victor W. Marek ◽  
Mirosław Truszczyński
Keyword(s):  
Positive Integer ◽  
Local Search ◽  
Ramsey Theory ◽  
Upper Bounds ◽  
General Purpose ◽  
Propositional Satisfiability ◽  
Lower And Upper Bounds ◽  
Algebraic Form ◽  
Sat Solvers ◽  
Structural Simplicity

In this paper we bring together the areas of combinatorics and propositional satisfiability. Many combinatorial theorems establish, often constructively, the existence of positive integer functions, without actually providing their closed algebraic form or tight lower and upper bounds. The area of Ramsey theory is especially rich in such results. Using the problem of computing van der Waerden numbers as an example, we show that these problems can be represented by parameterized propositional theories in such a way that decisions concerning their satisfiability determine the numbers (function) in question. We show that by using general-purpose complete and local-search techniques for testing propositional satisfiability, this approach becomes effective — competitive with specialized approaches. By following it, we were able to obtain several new results pertaining to the problem of computing van der Waerden numbers. We also note that due to their properties, especially their structural simplicity and computational hardness, propositional theories that arise in this research can be of use in development, testing and benchmarking of SAT solvers.

scholarly journals On the Growth of Some Functions Related to z(n)

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 876 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Positive Integer ◽  
Arithmetic Function ◽  
Fibonacci Sequence ◽  
Upper Bounds ◽  
Integer Solution ◽  
Lower And Upper Bounds ◽  
Positive Integer Solution

The order of appearance z : Z > 0 → Z > 0 is an arithmetic function related to the Fibonacci sequence ( F n ) n . This function is defined as the smallest positive integer solution of the congruence F k ≡ 0 ( mod n ) . In this paper, we shall provide lower and upper bounds for the functions ∑ n ≤ x z ( n ) / n , ∑ p ≤ x z ( p ) and ∑ p r ≤ x z ( p r ) .

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