the asymptotic behaviour of the sequence with general term $p_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $p_n-6/pi^2$ is found.

d|en d denote the number and the sum of exponential divisors of n, respectively. Properties of these functions were investigated by several authors, see [1], [2], [3], [5], [6], [8]. Two integers n,m > 1 have common exponential divisors iff they have the same prime factors and for n = ∏r i=1 p ai i , m = ∏r i=1 p bi i , ai, bi ≥ 1 (1 ≤ i ≤ r), the greatest common exponential divisor of n and m is

We call the function / described in the theorem a coprime mapping. Theorem 1 settles in the affirmative a conjecture of D. J. Newman. The special case when / = {JV + l.JV + 2, ...,2Af} was proved by D. E. Daykin and M. J. Baines [2]. V. Chvatal [1] established Newman's conjecture for each JV ^ 1002. We prove Theorem 1 constructively by giving an algorithm for the construction of a coprime mappi...

The asymptotic behaviour of the sequence with general term $P_n=(varphi(1)+varphi(2)+cdots+varphi(n))/(1+2+cdots+n)$, is studied which appears in the studying of coprime integers, and an explicit bound for the difference $P_n-6/pi^2$ is found.

We show, if p is prime, that the equation xn + yn = 2pz2 has no solutions in coprime integers x and y with |xy| ≥ 1 and n > p132p , and, if p 6= 7, the equation xn + yn = pz2 has no solutions in coprime integers x and y with |xy| ≥ 1 and n > p12p .

This paper addresses a problem recently raised by Laurent and Nogueira about inhomogeneous Diophantine approximation with coprime integers. A corollary of our main theorem is that for any irrational α ∈ R and for any γ ∈ R and > 0 there are infinitely many pairs of coprime integers m,n such that |nα−m− γ| ≤ 1/|n| . This improves upon previously known results, in which the exponent of approximat...

For g, n coprime integers, let `g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of `g(n) as n ≤ x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptot...

For g, n coprime integers, let `g(n) denote the multiplicative order of g modulo n. Motivated by a conjecture of Arnold, we study the average of `g(n) as n ≤ x ranges over integers coprime to g, and x tending to infinity. Assuming the Generalized Riemann Hypothesis, we show that this average is essentially as large as the average of the Carmichael lambda function. We also determine the asymptot...