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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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