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Summary  

We provide a general asymptotic formula which permits applications to sums like \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} $\sum_{x< n\le x+y} \big(d(n)\big)^2, \quad \sum_{x< n\le x+y} d(n^3),\quad \sum_{x< n\le x+y}\big(r(n)\big)^2, \quad \sum_{x< n\le x+y}r(n^3),$ \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} $d(n)$ \end{document} and \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} $r(n)$ \end{document} are the usual arithmetic functions (number of divisors, sums of two squares), and \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} $y$ \end{document} is small compared to~\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} $x$ \end{document}.

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Abstract  

For the number of integer solutions of the title equation, withW≤;x (x a large parameter), an asymptotics of the form Ax log x + Bx + O(x 1/2 (log x)3 (loglog x)2) is established. This is achieved in a general setting which furnishes applications to some other natural arithmetic functions.

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] Deza , E. , Varukhina , L. 2008 On mean values of some arithmetic functions in number fields Discrete Math. 308 4892 – 4899 10.1016/j.disc.2007.09.008 . [4] Heath-Brown , D. R

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For a large class of arithmetic functions f, it is possible to show that, given an arbitrary integer κ ≤ 2, the string of inequalities f(n + 1) < f(n + 2) < … < f(n + κ) holds for in-finitely many positive integers n. For other arithmetic functions f, such a property fails to hold even for κ = 3. We examine arithmetic functions from both classes. In particular, we show that there are only finitely many values of n satisfying σ2(n − 1) < σ2 < σ2(n + 1), where σ2(n) = ∑d|n d 2. On the other hand, we prove that for the function f(n) := ∑p|n p 2, we do have f(n − 1) < f(n) < f(n + 1) in finitely often.

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Abstract  

We give a new characterization of the logarithm as an additive arithmetical function.

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Abstract  

Let m and n be positive integers, and the M"bius function. And let S f(m,n) be the function 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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\Sigma _{d|(m,n)} d\mu (m/d)f(n/d)$$ \end{document}
, where f is an arithmetical function. We show that this function has many properties like the Ramanujan sum. Firstly we study the partial summation formula involving S f(m,n) and taking f=, we obtain the Dirichlet series with the coefficients S(m,n) and S(m,n)d(m). Moreover we show a certain property which is analogous to the orthogonality relation of the Ramanujan sums.

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пУсть — АлгЕБРА тАкИ х пОДМНОжЕстВ НАтУРА льНых ЧИсЕл, ЧтО кАжДОЕA А сИМптОтИЧЕскИ плОтНО, И ℰ( ) —лИНЕИНОЕ п РОстРАНстВО пРОстых ФУНкцИИ НА МНОжЕстВАх Иж . РАссМАтРИВАЕтсь тОп ОлОгИЧЕскОЕ жАМыкАН ИЕ ℒ*q( ) пРОстРАНстВА ℰ( ) ОтНОс ИтЕльНО пОлУНОРМы , ОпРЕДЕльЕМ ОИ сООтНОшЕНИЕМ

\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} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\bar M(|f|^q ): = \mathop {\lim }\limits_{x \to \infty } x^{ - 1} \mathop \Sigma \limits_{n\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } x} |f(n)|^q (1\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } q< \infty ).$$ \end{document}
пОкАжАНО, ЧтО Дль кАжД ОИf∃ℒ*q( ) сУЩЕстВУУт пРЕДЕльНОЕ РАспРЕДЕ лЕНИЕ, сРЕДНЕЕ жНАЧЕН ИЕ И МОМЕНты ВсЕх пОРьДк ОВr, 1≦rq, кРОМЕ тОгО, ДА Етсь хАРАктЕРИстИкА ДВОИ стВЕННОгО пРОстРАНстВА ДльL *q, ( ):=ℒ*q( )/N q( ), гДЕN q( ) — НУль-пРОстРАНстВ О ОтНОсИтЕльНО .

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