On a relation between the multidimensional Dirichlet divisor problem and bounds of the zeros of \(\zeta(s)\) (Q1809995)

For a positive integer \(k\), and \(x\geq 2\), let \[ R_{k}(x)=\sum_{1\leq n\leq x}\tau_{k}(n) - \text{ Res}_{s=1} \biggl(\zeta^{k}(s){x^{s}\over s}\biggr), \] where \(\tau_{k}(n)\) is the number of solutions in positive integers \(x_{1},\ldots,x_k \) of \(x_{1}\cdots x_{k}=n\). Dirichlet, who posed the problem of determining the order of growth of \(|R_{k}(x)|\) as \(x\to\infty\) for any \(k\geq 2\), showed that \[ |R_{k}(x)|\leq x^{1 - 1/k}(c\log x)^k , \] where \(c\) is an absolute constant. In this paper the author continues his study of the relationship between the size of \(|R_{k}(x)|\) and the zero-free region for \(\zeta(s)\). The main result is as follows. Suppose \(\alpha(y)\) is a nonincreasing function of \(y \geq 2\) with \(y^{-1} \leq \alpha(y) \leq {1\over 2}\), and for some \(c\geq 2\) and all \(k\geq 2, x\geq 2\) assume that \[ |R_{k}(x)|\leq x^{1-\alpha(k)}(c\log x)^{k} . \] Then \(\zeta(\sigma +it) \neq 0\) in \[ \sigma \geq 1 -c_{1}\alpha \Biggl({\log |t|\over \log\log |t|} \Biggr) (\log\log |t|)^{-1}, \quad |t|\geq e^{10} \] with some absolute constant \(c_1 > 0\). This improves an earlier result of the author [Funct. Approximatio, Comment. Math. 28, 131-140 (2000; Zbl 0977.11042)]. In this context, Dirichlet's bound yields the zero-free region of de la Vallée Poussin, and the author's estimate \[ |R_{k}(x)|\leq x^{1-ak^{-{2\over 3}}}(c\log x)^k \] [Izv. Akad. Nauk SSSR, Ser. Mat. 36, 475-483 (1972; Zbl 0239.10025)] gives Vinogradov's zero-free region. It is noted that under the same assumptions \[ \biggl|\sum_{n\leq a}n^{it}\biggr|\leq c_{1}a^{1-\beta}\log^{3}t, \] where \(10<a\leq t\) and \(\beta = {1\over 2}\alpha(\log t/\log\log t)\). Moreover it is proved that there is a sequence of \(y\to\infty\) for which \[ \alpha(y) \leq 8{\log\log y\over \log y}. \]