The multidimensional Dirichlet divisor problem and zero free regions for the Riemann zeta function (Q5933538)

Let \(d_k(n)\) denote the number of ways \(n\) can be represented as a product of \(k\) factors, and let NEWLINE\[NEWLINE \Delta_k(x) := \sum_{n\leq x}d_k(n) - xP_{k-1}(\log x) NEWLINE\]NEWLINE denote the error term in the (generalized or multidimensional) Dirichlet divisor problem, where \(P_{k-1}(y)\) is a polynomial of degree \(k-1\) in \(y\). From estimates on \(\zeta(\sigma + it)\) one can estimate \(\Delta_k(x)\) [see e.g., Chapter 13 of the reviewer's monograph ``The Riemann zeta-function'', John Wiley \& Sons (1985; Zbl 0556.10026) and his joint paper with \textit{M. Ouellet}, Acta Arith. 59, 241-253 (1989; Zbl 0619.10041)]. NEWLINENEWLINENEWLINEIn this work the author proceeds in the opposite direction, namely he starts from estimates for \(\Delta_k(x)\) and obtains bounds for \(\zeta(\sigma + it)\) and the closely related problem of the zero-free region for \(\zeta(s)\). He proves three theorems, too technical to be reproduced here. As a corollary of his results it follows that the known (uniform) bound NEWLINE\[NEWLINE \Delta_k(x) \ll x^{1-ck^{-2/3}}(C\log x)^k\quad(k\geq 2, C>0) NEWLINE\]NEWLINE implies the zero-free region (\(s = \sigma + it\)) NEWLINE\[NEWLINE \sigma \geq 1 - {c_1\over\log^{2/3}|t|\log\log|t|} \quad (|t|\geq {\text e}^2, c_1 > 0) NEWLINE\]NEWLINE for \(\zeta(s)\). The above region is, by a \(\log\log\)-factor, weaker than the strongest known zero-free region for \(\zeta(s)\) (see Chapter 6 of the reviewer's monograph, op. cit.).NEWLINENEWLINENEWLINEReviewer's remark: The zero-free regions under 2. and 3. on p. 137, e.g. NEWLINE\[NEWLINE \sigma \geq 1 - {c_2\over\log\log|t|(\log\log\log|t|)^\alpha}, NEWLINE\]NEWLINE are only conditional, and depend on hitherto unproved bounds for the function \(\alpha(k)\).