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Asymptotics of certain number-theoretic sums. (English. Russian original) Zbl 1205.11106
Math. Notes 83, No. 3, 432-436 (2008); translation from Mat. Zametki 83, No. 3, 472-476 (2008).
Let $$\theta$$ be a quadratic irrationality. For $$s>1$$, the two-sided following asymptotic estimates are known (see for instance [A. H. Kruse, Acta Arith. 12, 229–261 (1967; Zbl 0153.07001)] and [S. Haber and C. F. Osgood, Pac. J. Math. 31, 383–394 (1969; Zbl 0203.35602)]): $\sigma_1(N,\theta,s)=\sum_{n=1}^N \frac{1}{<n\theta>^s}\asymp N^s,\;\;\sigma_2(N,\theta,s)=\sum_{n=1}^N\frac{1}{<n\theta>^sn^s}\asymp ln\;N,$ where $$<t>$$ denotes the distance from $$t$$ to the nearest integer.
In the present paper the author solves the question of the asymptotics of a sum similar to $$\sigma_2(N,\theta,s)$$, namely of the sum $\sum_{n=1}^N\frac{1}{(N^{1-s}+<n\theta>^s)n^s},\;s>1,$ where $$\theta$$ is a quadratic irrationality such that $$\theta+\theta^\prime\in\mathbb Z$$ (with $$\theta^\prime$$ denoting the conjugate of $$\theta$$). The solution of this question is based on the knowledge of the asymptotic behaviour as $$h\to 0$$ of sums of the form $S(h,M,s)=\sum_{(0,0)\not= (x,y)\in M}\frac{1}{(h^s+|x|^s)(h^s+|y|^s)}, \;s>1,$ where $$M$$ is the lattice in the plane $$\mathbb R^2$$ representing some complete module in a real quadratic field (for the method see [N. N. Osipov, Vychisl. Tekhnol. 11, Spec. Iss. 4, 81–89 (2006; Zbl 1201.11091)]).
Suppose here that $$a\theta^2+b\theta+c=0$$ where $$a,b,c$$ are coprime integers with $$a>0$$. In the quadratic field $$\mathfrak R=\mathbb Q(\theta)$$ consider the module $$\mathfrak M:=\{1,-\theta\}=\{m-n\theta : (m,n)\in\mathbb Z^2\}$$. Let $$\mathfrak M_1=a\mathfrak M=\{a,-a\theta\}$$ similar to the module $$\mathfrak M$$ contained in the ring $$\mathfrak D=\{1,a\theta\}$$. For an arbitrary natural number $$c$$, let $$Q(c)$$ defined by: $$\mu_j\in \mathfrak M_1, \;j=1,\dots, Q(c)$$ is the fixed collection of pairwise non associated solutions of the norm equation $$|N_{\mathfrak R/\mathbb Q}(\mu)|=c$$.
The author has proved: Theorem: If $$\theta+\theta^\prime\in\mathbb Z$$ then, as $$h\to 0$$, the following estimate holds: $\sum_{n\geq 1}\frac{1}{(h^s+<n\theta>^s)n^s}= \frac{a^{2s}|\theta-\theta^\prime|^s}{ln\;\varepsilon}\sum_{c\geq 1}\frac{Q(c)}{c^s} ln\;h^{-1}+O(1),$ where $$\varepsilon>1$$ is the fundamental unit of the order $$\mathfrak D=\{1,a\theta\}$$.
Corollary: If $$\theta+\theta^\prime\in\mathbb Z$$ then, as $$N\to\infty$$, the following estimate holds: $\sum_{n=1}^N\frac{1}{(N^{1-s}+<n\theta>^s)n^s}= \frac{a^{2s}|\theta-\theta^\prime|^s}{ln\;\varepsilon}\sum_{c\geq 1}\frac{Q(c)}{c^s} \nu \;ln N+O(1),$ where $$\nu=1-1/s$$ and $$\varepsilon>1$$ is the fundamental unit of the order $$\mathfrak D=\{1,a\theta\}$$.
##### MSC:
 11N37 Asymptotic results on arithmetic functions 11N56 Rate of growth of arithmetic functions
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##### References:
 [1] W. Schmidt, Diophantine Approximation (Springer-Verlag, Heidelberg, 1980; Mir, Moscow, 1983). · Zbl 0421.10019 [2] A. H. Kruse, Acta Arith. 12, 229 (1967). · Zbl 0153.07001 [3] S. Haber and C. Osgood, Pacific J. Math. 31, 383 (1969). and · Zbl 0203.35602 [4] N. N. Osipov, Computational Techniques 11 (Special Issue), 81 (2006). [5] Z. I. Borevich and I. R. Shafarevich, Number Theory (Nauka, Moscow, 1985) [in Russian].
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