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Abstract  

We shall investigate several properties of the integral

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int_1^\infty {t^{ - \theta } \Delta _k \left( t \right) log^j t dt}$$ \end{document}
with a natural number k, a non-negative integer j and a complex variable θ, where Δk(x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the “elementary methods” and the “elementary formulas” to derive convergence properties and explicit representations of this integral with respect to θ for k = 2.

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Abstract

Deza and Varukhina [3] established asymptotic formulae for some arithmetic functions in quadratic and cyclotomic fields. We generalize their results to any Galois extension of the rational field. During this process we rectify the main terms in their asymptotic formulae.

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Abstract

For the Riemann zeta-function we present an asymptotic formula of a shifted fourth moment in an unbounded shift range along the critical line.

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Abstract

We show, conditional on a uniform version of the prime k-tuples conjecture, that there are x/(log x)1+o(1) numbers not exceeding x common to the ranges of ϕ and σ. Here ϕ is Euler’s totient function and σ is the sum-of-divisors function.

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Abstract

We study two general divisor problems related to Hecke eigenvalues of classical holomorphic cusp forms, which have been considered by Fomenko, and by Kanemitsu, Sankaranarayanan and Tanigawa respectively. We improve previous results.

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Abstract

Let f(x) be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate log lcm (f(1),…,f(n))∼An as n→∞, where A is a constant depending on f.

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Abstract  

We study the irrational factor function I(n) introduced by Atanassov and defined by

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$I(n) = \prod\nolimits_{\nu = 1}^k {p_\nu ^{1/\alpha _\nu } }$$ \end{document}
, where
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$n = \prod\nolimits_{\nu = 1}^k {p_\nu ^{\alpha _\nu } }$$ \end{document}
is the prime factorization of n. We show that the sequence {G(n)/n}n≧1, where G(n) = Πν=1 n I(ν)1/n, is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n).

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Abstract  

In this article we consider three problems: 1. The asymptotic behaviour of the quadratic moment of the exponential divisor function. 2. The distribution of powerful integers of type 4. 3. The average number of direct factors of a finite Abelian group. We prove new estimates for the error terms in the asymptotic representations. For this purpose new estimates in the general four-dimensional divisor problem are needed.

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Abstract  

On path partitions of the divisor graph. Let D(x) be the graph with vertices {1, 2, ..., ⌊x⌋} whose edges come from the division relation, and let D(x, y) be the subgraph restricted to the integers with prime factors less than or equal to y. We give sufficient conditions on x and y for the graph D(x, y) to be Hamiltonian. We deduce an asymptotic formula for the number of paths in D(x) needed to partition the set of vertices {1, 2, ..., ⌊x⌋}.

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