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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
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
Some Arithmetic Functions Involving Exponential Divisors
"... 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. 1 Introduction and ..."
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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. 1 Introduction and
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], H. L. Mo ..."
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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^+£).