Improvement of an estimate of H. Müller involving the order of \(2 \pmod u\). II
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Publication:2501158
DOI10.1007/S00013-006-1704-ZzbMATH Open1133.11055arXivmath/0507596OpenAlexW2076696288MaRDI QIDQ2501158
Publication date: 4 September 2006
Published in: Archiv der Mathematik (Search for Journal in Brave)
Abstract: Let m>=1 be an arbitrary fixed integer and let N_m(x) count the number of odd integers u<=x such that the order of 2 modulo u is not divisible by m. In case m is prime estimates for N_m(x) were given by H. Mueller that were subsequently sharpened into an asymptotic estimate by the present author. Mueller on his turn extended the author's result to the case where m is a prime power and gave bounds in the case m is not a prime power. Here an asymptotic for N_m(x) is derived that is valid for all integers m. This asymptotic would easily have followed from Mueller's approach were it not for the fact that a certain Diophantine equation has non-trivial solutions. All solutions of this equation are determined. We also generalize to other base numbers than 2. For a very sparse set of these numbers Mueller's approach does work.
Full work available at URL: https://arxiv.org/abs/math/0507596
Asymptotic results on arithmetic functions (11N37) Other results on the distribution of values or the characterization of arithmetic functions (11N64) Rate of growth of arithmetic functions (11N56)
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