On the squarefree and squarefull numbers. (English) Zbl 1089.11052

For \(p\) prime it is shown that there are \[ cxp\varphi(p-1)/(p^2-1)+O(p^{9/44+ \varepsilon}) \] square-free primitive roots up to \(x\), and \[ c'x^{1/2}p\varphi (p-1)/(p^2-1)+O(p^{9/44+\varepsilon}x^{1/4 +\varepsilon}) \] square-full primitive roots up to \(x\). Here \(c\) and \(c'\) are explicitly given constants. It is also shown that there are \(3\pi^{-2}x +O(p^{9/44+\varepsilon}x^{1/2+\varepsilon})\) square-free quadratic residues up to \(x\). The proofs depend on the bound \[ \int^T_0\frac {\bigl| L(1/2+it,\chi)\bigr|}{1+t}dt\ll p^{9/44+ \varepsilon} T^\varepsilon \] for a non-principal character \(\chi\) modulo \(p\). This is proved using a mean-value theorem for \(L(s,\chi)\) due to M. Katsurada and K. Matsumoto [Number theory and its applications (Dordrecht: Kluwer), Dev. Math. 2, 199–229 (1999; Zbl 0966.11036)].


11N25 Distribution of integers with specified multiplicative constraints
11M06 \(\zeta (s)\) and \(L(s, \chi)\)


Zbl 0966.11036
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