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The distribution of powerful integers of type 4. (English) Zbl 0679.10035

A natural number \(n\) is said to be powerful of type \(k\) if \(p^ k\) divides \(n\) whenever \(p\) is a prime divisor of \(n\). The asymptotic behaviour of the counting function for such numbers was first considered by P. Erdős and G. Szekeres [Acta Sci. Math. 7, 95–102 (1934; Zbl 0010.29402)]. For \(k=4\), let \(\Delta_ 4(x)\) be the associated error term, and let \[ \lambda_ 4=\inf \{\rho_ 4: \Delta_ 4(x)\ll x^{\rho_ 4}\}. \] P. T. Bateman and E. Grosswald [Ill. J. Math. 2, 88–98 (1958; Zbl 0079.07104)] first obtained the estimate \(\lambda_ 4\leq 1/6\), and a number of improvements have appeared since; for example, A. Ivić and the reviewer [Ill. J. Math. 26, 576–590 (1982; Zbl 0484.10024)] and E. Krätzel [Elementary and analytic theory of numbers, Banach Cent. Publ. 17, 337–369 (1985; Zbl 0595.10035)] gave \(\lambda_ 4\leq 3091/25981\) and \(\lambda_ 4\leq 21/187=0.1122...\) respectively. The problem is reduced to the estimation of a certain three-dimensional exponential sum, and the author proves that \(\lambda_ 4\leq 35/316=0.1107....\).

MSC:

11N25 Distribution of integers with specified multiplicative constraints
11N37 Asymptotic results on arithmetic functions
11L07 Estimates on exponential sums
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References:

[1] Bateman, P., Grosswald, E.: On a theorem of Erd?s and Szekeres. Illinois J. Math.2, 88-98 (1958). · Zbl 0079.07104
[2] Erd?s, P., Szekeres, G.: ?ber die Anzahl der Abelschen Gruppen gegebener Ordnung und ?ber ein verwandtes zahlentheoretisches Problem. Acta Sci. Math. (Szeged)7, 95-102 (1935). · JFM 60.0893.02
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[8] Kr?tzel, E.: Zweifache Exponentialsummen und dreidimensionale Gitterpunktprobleme. In: Elementary Analytic Theory of Numbers. (H. Iwaniec ed.) Warsaw: PWN-Polish Scientific Publ. pp. 337-369 (1985).
[9] Kr?tzel, E.: Lattice Points. Dordrecht-Boston-London: Kluwer Academic Publ. 1988.
[10] Kr?tzel, E.: On the average number of direct factors of a finite Abelian group. Acta Arith., to appear.
[11] Kr?tzel, E.: The distribution of powerful integers of type 4. To appear.
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[13] Vogts, M.: Many-dimensional generalized divisor problems. Math. Nachr.124, 103-121 (1985). · Zbl 0595.10034 · doi:10.1002/mana.19851240108
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