The multidimensional Dirichlet divisor problem and the Carlson abscissa and exponent (Q960651)

The author announces (without proofs) several new results involving the general (multidimensional) Dirichlet divisor problem, which concerns the function \[ \Delta_k(x) = \sum_{n\leq x}d_k(n) - xP_{k-1}(\log x), \] where \(d_k(n)\) is the divisor function generated by \(\zeta^k(s)\), and \(P_{k-1}(y)\) is a well-known polynomial of degree \(k-1\) in \(y\), whose coefficients depend on \(k\). Let, as usual, \(\alpha_k\) and \(\beta_k\) denote the least numbers \(a\) and \(b\) for which one has \[ \Delta_k(x) \ll x^a,\qquad \int_1^x\Delta_k^2(y)dy \ll x^{1+2b}, \] respectively. The main results are the new bounds for \(\alpha_k, \beta_k\) when \(k\) is large. For \(k\geq 93\) one has \[ \alpha_k \leq 1 - \left({2\over3a(k-2k_1)}\right)^{2/3}, \quad k_1 = 79.95, \leqno(1) \] and \(a\) is the constant for which \[ \zeta(\sigma + it) \ll t^{a(1-\sigma)^{2/3}}\log t, \qquad(t>1,\, 0.9 < \sigma\leq 1,\, 1\leq a \leq 20). \] The above bound sharpens the result of \textit{M. Ouellet} and the reviewer [Acta Arith. 52, No. 3, 241--253 (1989; Zbl 0619.10041)]. For \(k\geq 93, k_1 = 79.95\) one has \[ \beta_k \leq 1 - \left({b\over 3ak_3}\right)^{2/3}, \quad k=k-k_1,\, a = 4.45,\, b = 2.5. \] This is again sharper than the corresponding result of Ivić--Ouellet [op. cit.]. Reviewer's remark: The bound (1) is meaningful for \(k > 2k_1\), that is, for \(k\geq 160\).