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On the general asymmetric divisor problem. (English) Zbl 0854.11048
For an integer $$p\geq 2$$ and fixed natural numbers $$a_1\leq a_2\leq \dots \leq a_p$$, let $$d(a_1, \dots, a_p; n)$$ denote the general asymmetric divisor function $d(a_1, \dots, a_p; n)= \#\{( m_1, \dots, m_p)\in \mathbb{N}^p: m_1^{a_1} \dots m_p^{a_p}= n\} \qquad (n\in \mathbb{N}).$ To describe its average order one is interested in asymptotic formulas for its Dirichlet summatory function of the form $$\sum_{n\leq x} d(a_1, \dots, a_p; n)= H(a_1, \dots, a_p; x)+ \Delta (a_1, \dots, a_p; n)$$ with $$x$$ a large real variable, where the main term $$H(a_1, \dots, a_p; x)$$ is a certain sum over residues of the generating function. The present state-of-the-art concerning the sharpest and most general upper estimates for the error term $$\Delta (a_1, \dots, a_p; x)$$ can be found in E. Krätzel [Abh. Math. Semin. Univ. Hamb. 62, 191-206 (1992; Zbl 0776.11057)]. This paper aims at lower bounds for $$\Delta (a_1, \dots, a_p; x)$$. Its ultimate goal (Theorem 2) is to prove that \begin{aligned} \Delta (a_1, \dots, a_p; x) &= \Omega_\pm (x^\theta (\log x)^{a_1 \theta} (\log \log x)^{p- 1}) \quad \text{ for } p\geq 4,\\ \text{and} \Delta (a_1, \dots, a_p; x) &= \Omega_+ (x^\theta (\log x)^{a_1 \theta} (\log \log x)^{p- 1}) \quad \text{ for } p=2, 3, \end{aligned} where $$\theta= (p- 1)/ (a_1+ \dots+ a_p)$$. Theorem 3 provides a quantitative refinement which shows how quickly the oscillations happen if $$p\geq 4$$. As a by-result (Theorem 1), which however might have some interest of its own, the author derives a representation for the Riemann-Liouville integral of order greater than $${{p-1} \over 2}$$ by an absolutely convergent series over generalized cylinder functions for which in turn sharp asymptotic expansions are provided.
The argument is based on methods of K. Chandrasekharan and R. Narasimhan [Math. Ann. 152, 30-64 (1963; Zbl 0116.27001)], B. C. Berndt [J. Number Theory 3, 184-203 (1971; Zbl 0216.31303) and 288-305 (1971; Zbl 0219.10050)], and J. Hafner [J. Number Theory 15, 36-76 (1982; Zbl 0495.10027)]. As the author remarks, it works best if the $$a_i$$’s are all “approximately equal”: For “strongly asymmetric” cases (e.g., $$(a_1, a_2, a_3)= (1, 2, 3))$$, the reviewer’s paper [J. Number Theory 27, 73-91 (1987; Zbl 0619.10040)] contains better $$\Omega$$ (though not $$\Omega_\pm$$) bounds. The interested reader is also referred to a forthcoming paper of M. Kühleitner [submitted for publication] where the results are sharpened by log log-factors on the basis of a refined version of Hafner’s method.
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