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On the asymmetric divisor problem with congruence conditions. (English) Zbl 0852.11052
For $$N= p+q\geq 2$$, $$p,q\in \mathbb{N}$$ and fixed natural numbers $$a_1, \dots, a_p$$, $$a_{p+1}= b_1, \dots, a_{p+q}= b_q$$, let $$d^* (n)$$ denote the number of representations $$n= u_1^{a_1} \dots u_N^{a_N}$$ with $$u_j\equiv \ell_j \pmod {m_j}$$ $$(j=1, \dots, p)$$, where $$\ell_j, m_j\in \mathbb{N}$$, $$\ell_j< m_j$$. The author considers $$E(x)$$, the error term in the asymptotic formula for the summatory function of $$d^* (n)$$. His main objective is the lower bound for $$E(x)$$, and he proves two theorems, the formulation of which is too complicated to be reproduced here. In the first theorem an absolutely convergent series representation (which is quite explicit) is provided for the function $$\int^x_0 (x- u)^{m-1} E(u) du$$ if $$m> (N- 1)/2$$, $$m\in \mathbb{N}$$. This is achieved by the complex integration technique. For the second result, the author combines the classical technique of Szegö and Walfisz with the more recent method of J. L. Hafner [J. Number Theory 28, 240-257 (1988; Zbl 0635.10037)] for obtaining $$\Omega$$-results. He obtains an $$\Omega_\pm$$-result for $$E(x)$$, where the exponents of $$x$$ and (iterations of) $$\log x$$ depend on the parameters in the asymmetric divisor problem in an explicit way. The proof of Theorem 2 uses, among other things, the asymptotic expansion of Theorem 1.