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Published in:
2016 | OriginalPaper | Chapter
On Error Sum Functions for Approximations with Arithmetic Conditions
Author : Carsten Elsner
Published in: From Arithmetic to Zeta-Functions
Publisher: Springer International Publishing
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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 \(0 \leq l <k\). The functions \(\mathcal{E}_{k,l}\) are Riemann-integrable on \([0,1]\), so that the integrals \(\int _{0}^{1}\mathcal{E}_{k,l}(\alpha )\,d\alpha\) exist as the arithmetical means of the functions \(\mathcal{E}_{k,l}\) on \([0,1]\). We express these integrals by multiple sums on rational terms and prove upper and lower bounds. In the case when \(l\) vanishes (i.e. \(k\) divides \(q_{m}\)) and when the smallest prime divisor \(p_{1}\) of \(k = p_{1}^{a_{1}}p_{2}^{a_{2}}\cdots p_{t}^{a_{t}}\) satisfies \(p_{1}> k^{\varepsilon }\) 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 \(a \geq 1\).