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Some new estimates in the Dirichlet divisor problem. (English) Zbl 0619.10041

For a fixed integer \(k\geq 2\), let as usual \[ \Delta _ k(x)=\sum _{n\leq x}d_ k(n)-_{s=1} x^ s\zeta ^ k(s)s^{-1},\quad \zeta ^ k(s)=\sum ^{\infty}_{n=1}d_ k(n)n^{-s}\quad (Re s>1), \] and let \(\alpha _ k\) and \(\beta _ k\) denote the infima of positive numbers \(a_ k\) and \(b_ k\) for which \[ \Delta _ k(x)\ll x^{a_ k},\quad \int ^{x}_{1}\Delta ^ 2_ k(y) dy\ll x^{1+2b_ k}. \] Using complex integration and techniques from the theory of the Riemann zeta-function, several new upper bounds for \(\alpha _ k\) and \(\beta _ k\) are derived [see Ch. 13 of the first author’s monograph ”The Riemann zeta-function” (Wiley, New York 1985; Zbl 0556.10026) for a comprehensive account on problems involving \(\Delta _ k(x)]\). In particular, if for some \(D>0\) \[ \zeta (\sigma +it)\ll t^{D(1-\sigma)^{3/2}}(\log t)^{2/3}\quad (t\geq t_ 0,\quad \leq \sigma \leq 1) \] holds, then \[ \alpha _ k\leq 1-\frac{1}{3}\cdot 2^{2/3}(Dk)^{-2/3},\quad \beta _ k\leq 1-\frac{2}{3}(Dk)^{-2/3}, \] and the methods of the paper yield the sharpest known upper bounds for \(\alpha _ k\) when \(k\geq 10\).

MSC:

11N37 Asymptotic results on arithmetic functions
11M06 \(\zeta (s)\) and \(L(s, \chi)\)

Citations:

Zbl 0556.10026
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