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Arithmetical functions involving exponential divisors: note on two papers by L. Tóth. (English) Zbl 1240.11116

Summary: Asymptotic estimates of L. Tóth [Ann. Univ. Sci. Budapest Sect. Comp. 24, 285–294 (2004; Zbl 1108.11071)], [Ann. Univ. Sci. Budapest Sect. Comp. 27, 155–166 (2007; Zbl 1164.11055)] on the summatory functions of three arithmetical functions involving exponential divisors are improved. For two of them the improvement is on the upper bound of the size of the remainder term (\(O\)-estimate), and is reached by appealing to lattice points estimates using exponent pairs due to E. Krätzel [Lattice points, Kluwer Academic Publishers, Berlin (1988; Zbl 0675.10031)], and by having as well a closer look at the first terms of the generating Dirichlet series. For the third one, a lower bound on the size of the remainder term (\(\Omega\)-estimate) is replaced by two-sided oscillation (\(\Omega_\pm\)-estimate), by appealing to a method of Y.-F. S. Pétermann and J. Wu [Acta Math. Hung. 77, 159–175 (1997; Zbl 0902.11037)].

MSC:

11N37 Asymptotic results on arithmetic functions

Online Encyclopedia of Integer Sequences:

Exponentially squarefree numbers.