The distribution of powerful integers of type 4

Abstract

LetN ₄(x) denote the number of powerful integers of type 4 not exceedingx. ForN ₄(x) one knows the following asymptotic representation

$$N_4 (x) = \sum\limits_{v = 4}^7 {\gamma _{v,4} x^{1/v} + \Delta _4 (x)} $$

where

$$\gamma _{v,4} = \mathop {\operatorname{Re} s}\limits_{s = 1/v} \frac{1}{s}F_4 (s), F_4 (s) = \prod\limits_p {\left( {1 + \frac{{p^{ - 4s} }}{{1 - p^{ - s} }}} \right)} $$

and Δ(x) is the remainder term. Using two different methods to estimate a special three-dimensional exponential sum we prove for$\lambda _4 = \inf \{ \varrho _4 :\Delta _4 (x)<< x^{\varrho 4} \}$ the better result$\lambda _4 \leqslant \frac{{35}}{{316}}$.