On a divisor problem in arithmetic progressions
Let d(\(\ell,K;n)\) denote the number of divisors of n which are congruent to \(\ell (mod K)\), \(1\leq \ell \leq K\). Put \[ \sum_{n\leq x}d(\ell,K;n)=(x/K)\log x+(\gamma (\ell,K)+(1/K)(\gamma -1))x+\Delta (\ell,K;x) \] where \(\gamma\) denotes the Euler constant and \(\gamma (\ell,K)=\lim_{x\to \infty}(\sum_{n\leq x,n\equiv \ell (mod K)}1/n- (1/K)\log x).\) In this paper, the author observes that the ``discrete Hardy-Littlewood method'' may be applied to prove that \(\Delta (\ell,K;x)=O((x/K)^{\theta})\), for any \(\theta >7/22\), uniformly in \(\ell \leq K\leq x\) and the main result of the paper is a lower estimate for \(\Delta\) (\(\ell,K;x)\). This depends on the two roots \(\lambda_ 1\) and \(\lambda_ 2\), in the unit interval of the transcendental equation \(-Log(2 \sin \pi \lambda)+\pi \lambda -\lambda /2=0.\) Reviewer's remark: Recently \textit{H. Menzer} and the author considered a more general problem which appeared in Manuscr. Math. 64, No.1, 107-119 (1989; Zbl 0674.10037)].
generalization of Dirichlet's divisor problem
scientific article; zbMATH DE number 4110800