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PRIMITIVE LATTICE POINTS IN STARLIKE PLANAR SETS
"... This article is concerned with the number BD(x) of integer points with relative prime coordinates in √ x D, where x is a large real variable and D is a starlike set in the Euclidean plane. Assuming the truth of the Riemann Hypothesis, we establish an asymptotic formula for BD(x). Applications to cer ..."
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This article is concerned with the number BD(x) of integer points with relative prime coordinates in √ x D, where x is a large real variable and D is a starlike set in the Euclidean plane. Assuming the truth of the Riemann Hypothesis, we establish an asymptotic formula for BD(x). Applications to certain special geometric and arithmetic problems are discussed. 1. Introduction. Let D denote a subset of R2 which is starlike with respect to the origin, i.e., if u ∈ R2 belongs to D, automatically λu ∈Dfor 0 <λ<1. The distance function F of D is defined by
CONDITIONAL ASYMPTOTIC FORMULAE FOR A CLASS OF ARITHMETIC FUNCTIONS
"... ABSTRACT. Under the assumption of Riemann's Hypothesis, a general asymptotic formula for sums ^ < a{n) is established which applies (e.g.) to arithmetic functions a(n) defined by a kind of convolution of the Möbius function with some divisor function. 1. Introduction. In a recent paper [5], ..."
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ABSTRACT. Under the assumption of Riemann's Hypothesis, a general asymptotic formula for sums ^ < a{n) is established which applies (e.g.) to arithmetic functions a(n) defined by a kind of convolution of the Möbius function with some divisor function. 1. Introduction. In a recent paper [5], H. L. Montgomery and R. C. Vaughan have investigated the distribution of fcfree integers under the assumption of the Riemann Hypothesis (henceforth quoted as RH). By a simple but ingenious new idea they succeeded in improving the classical error term 0(x2^2k+1^+e) (due to A. Axer [1]) in the formula for the number of fcfree integers < x to 0(x1^k+1^+£).
Some Arithmetic Functions Involving Exponential Divisors
, 2010
"... In this paper we study several arithmetic functions connected with the exponential divisors of integers. We establish some asymptotic formulas under the Riemann hypothesis, which improve previous results. We also prove some asymptotic lower bounds. ..."
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In this paper we study several arithmetic functions connected with the exponential divisors of integers. We establish some asymptotic formulas under the Riemann hypothesis, which improve previous results. We also prove some asymptotic lower bounds.
Sieve Methods
"... Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after th ..."
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Sieve methods have had a long and fruitful history. The sieve of Eratosthenes (around 3rd century B.C.) was a device to generate prime numbers. Later Legendre used it in his studies of the prime number counting function π(x). Sieve methods bloomed and became a topic of intense investigation after the pioneering work of Viggo Brun (see [Bru16],[Bru19], [Bru22]). Using his formulation of the sieve Brun proved, that the sum
Sieve Methods
"... called undeniable signature schemes require prime numbers of the form 2p 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their i ..."
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called undeniable signature schemes require prime numbers of the form 2p 1 such that p is also prime. Sieve methods can yield valuable clues about these distributions and hence allow us to bound the running times of these algorithms. In this treatise we survey the major sieve methods and their important applications in number theory. We apply sieves to study the distribution of squarefree numbers, smooth numbers, and prime numbers. The first chapter is a discussion of the basic sieve formulation of Legendre. We show that the distribution of squarefree numbers can be deduced using a squarefree sieve 1 . We give an account of improvements in the error term of this distribution, using known results regarding the Riemann Zeta function. The second chapter deals with Brun's Combinatorial sieve as presented in the modern language of [HR74]. We apply the general sieve to give a simpler
Sums of two relatively prime cubes
"... Let V (x) be the number of solutions (u, v) in Z2 of u3 + v3 ≤ x, (u, v) = 1, and let ..."
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Let V (x) be the number of solutions (u, v) in Z2 of u3 + v3 ≤ x, (u, v) = 1, and let
Primitive lattice points in planar domains
"... Let D be a compact convex set in R2, containing 0 as an interior point, having a smooth boundary curve C with nowhere vanishing curvature. How many primitive lattice points (m,n) (m ∈ Z, n ∈ Z, m, n coprime) are in√ xD for large x? If we write AD(x) for the number of such primitive points, ..."
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Let D be a compact convex set in R2, containing 0 as an interior point, having a smooth boundary curve C with nowhere vanishing curvature. How many primitive lattice points (m,n) (m ∈ Z, n ∈ Z, m, n coprime) are in√ xD for large x? If we write AD(x) for the number of such primitive points,