A brief contribution to the question of the divisor problem
Let \(\alpha_ k\), as usual, denote the optimal exponent in the generalized divisor problem, so that \[ \sum_{n \leq x} d_ k(n) = x p_ k (\log x) + O(x^{\alpha_ k + \varepsilon}). \] It was shown by \textit{A. A. Karatsuba} [Izv. Akad. Nauk SSSR, Ser. Mat. 36, 475-483 (1972; Zbl 0239.10025)] that \(\alpha_ k \leq 1-ck^{-2/3}\) \((k \geq k_ 0)\) with \(c = {1 \over 2} (2D)^{-2/3}\), where \(D\) is any constant for which the estimate \[ \zeta (\sigma + it) \ll t^{D(1-\sigma)^{3/2}} \log t\quad(t \geq 2,\quad {\textstyle{1 \over 2}} \leq \sigma \leq 1) \] holds. The main result of the paper is that any \(c<{1 \over 2} D^{-2/3}\) is admissible. Unfortunately this same observation has already been made by \textit{A. Ivić} [The Riemann zeta-function, p. 356 (Wiley, New York) (1985; Zbl 0556.10026)]. Indeed it appears to the reviewer that any \(c<({3 \over 2} D)^{-2/3}\) must be allowable. To prove this, one uses Theorem 7.9(A) of \textit{E. C. Titchmarsh} [The theory of the Riemann zeta-function (Oxford) (1986; Zbl 0601.10026)] to establish the bound \[ \sigma_ m \leq 1 - (3D + \varepsilon)^{-2/3} m^{-2/3} (m \geq m_ \varepsilon) \] (in Titchmarsh's notation), using induction on \(m\). One then has only to observe that \(\alpha_{2m}\), \(\alpha_{2m-1} \leq \sigma_ m\).
scientific article; zbMATH DE number 567925