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Summary  

For PF 2[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of  N (cf. [9]) such that Σn 0 p(A,n)z n  ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ d |n, d A d, namely, the periodicity of the sequences (σ(A,2k n) mod 2k +1)n 1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2k n) mod 2k +1 will be established, from which it will be shown that the weight σ(A1,2k z i) mod 2k +1    on the orbit \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} $z_i$ \end{document} is moved on some other orbit z j when A1 is replaced by A2 with A1= A(P 1) and A2= A(P 2) P 1 and P 2 being irreducible in F 2[z]  of the same odd order.

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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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We give upper bounds for sums of multiplicative characters modulo an integer q ≧ 2 with the Euler function ϕ ( n ) and with the shifted largest prime divisor P ( n ) + a of integers nx .

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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

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  

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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