On Error Sum Functions for Approximations with Arithmetic Conditions

Carsten Elsner ()
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Carsten Elsner: University of Applied Sciences, Fachhochschule für die Wirtschaft

A chapter in From Arithmetic to Zeta-Functions, 2016, pp 121-140 from Springer

Abstract: Abstract Let $$\mathcal{E}_{k,l}(\alpha ) =\sum _{q_{m}\equiv l\pmod k}\vert q_{m}\alpha - p_{m}\vert$$ be error sum functions formed by convergents $$p_{m}/q_{m}$$ $$(m \geq 0)$$ of a real number $$\alpha$$ satisfying the arithmetical condition $$q_{m} \equiv l\pmod k$$ with $$0 \leq l k^{\varepsilon }$$ for some positive real number $$\varepsilon$$ , we have found an asymptotic expansion in terms of $$k$$ , namely $$\int _{0}^{1}\mathcal{E}_{k,0}(\alpha )\,d\alpha =\zeta (2)\big(2\zeta (3)k^{2}\big)^{-1} + \mathcal{O}\big(3^{t}k^{-2-\varepsilon }\big)$$ . This result includes all integers $$k$$ which are of the form $$k = p^{a}$$ for primes $$p$$ and integers $$a \geq 1$$ .

Keywords: Approximation with arithmetical conditions; Continued fractions; Convergents; Error sum functions; Primary 11J04; Secondary 11J70 (search for similar items in EconPapers)
Date: 2016
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DOI: 10.1007/978-3-319-28203-9_9

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