On Error Sum Functions for Approximations with Arithmetic Conditions

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 $a \geq 1$.

Dedicated to the memory of Professor Wolfgang Schwarz