The divisor function on residue classes. III
Let \(d(n)\) denote the number of positive divisors of \(n\), and let \[ A(x,q,a)=\sum_{\substack{n\leqslant x\\ n\equiv\, a\, \pmod q}}d(n). \] The authors of the paper derive new estimates for the following second moment associated with \(A(x,q,a)\): \[ \sum_{q\leqslant Q}\sum_{a=1}^{q}\big|A(x,q,a)-\varrho_R(x,q,a)\big|^2, \] where \(\varrho_R(x,q,a)\) is an approximating function \[ \varrho_R(x,q,a)=\sum_{\substack{n\leqslant x\\ n\equiv\, a\, \pmod q}}\sum_{r\leqslant R}\frac{c_r(n)}{r}\big(\log\frac{n}{r^2}+2\gamma\big) \] with the Ramanujan sum \[ c_r(n)=\sum_{\substack{b=1\\ (b,r)=1}}^r\mathrm{e}^{2\pi bn\mathrm{i}/r}, \] and the Euler constant \(\gamma\). The paper is the continuation of the authors' works [Part II, the authors, Acta Arith. 182, No. 2, 133--181 (2018; Zbl 1439.11255)] and [Part I, the authors, ibid. 168, No. 4, 369--381 (2015; Zbl 1325.11100)].
scientific article; zbMATH DE number 7434822
scientific article