The asymptotic number of integral cubic polynomials with bounded heights and discriminants

Kaliada D, Götze F, Kukso O (2014)
Lithuanian Mathematical Journal 54(2): 150-165.

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Zeitschriftenaufsatz | Veröffentlicht | Englisch
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Abstract / Bemerkung
Let P denote a cubic integral polynomial, and let D(P) and H(P) denote the discriminant and height of P, respectively. Let N(Q,X) be the number of cubic integral polynomials P such that H(P) a parts per thousand currency sign Q and |D(P)| a parts per thousand currency sign X. We obtain an asymptotic formula of N(Q,X) for Q (14/5) a parts per thousand(a) X a parts per thousand(a) Q (4) and Q -> +a. Using this result, for 0 a parts per thousand currency sign eta a parts per thousand currency sign 9/10, we find the asymptotic value of where the sum is taken over irreducible integral polynomials and Q -> +a. This improves upon a result of Davenport, who dealt with the case eta = 0.
Erscheinungsjahr
Zeitschriftentitel
Lithuanian Mathematical Journal
Band
54
Ausgabe
2
Seite(n)
150-165
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Kaliada D, Götze F, Kukso O. The asymptotic number of integral cubic polynomials with bounded heights and discriminants. Lithuanian Mathematical Journal. 2014;54(2):150-165.
Kaliada, D., Götze, F., & Kukso, O. (2014). The asymptotic number of integral cubic polynomials with bounded heights and discriminants. Lithuanian Mathematical Journal, 54(2), 150-165. doi:10.1007/s10986-014-9234-z
Kaliada, D., Götze, F., and Kukso, O. (2014). The asymptotic number of integral cubic polynomials with bounded heights and discriminants. Lithuanian Mathematical Journal 54, 150-165.
Kaliada, D., Götze, F., & Kukso, O., 2014. The asymptotic number of integral cubic polynomials with bounded heights and discriminants. Lithuanian Mathematical Journal, 54(2), p 150-165.
D. Kaliada, F. Götze, and O. Kukso, “The asymptotic number of integral cubic polynomials with bounded heights and discriminants”, Lithuanian Mathematical Journal, vol. 54, 2014, pp. 150-165.
Kaliada, D., Götze, F., Kukso, O.: The asymptotic number of integral cubic polynomials with bounded heights and discriminants. Lithuanian Mathematical Journal. 54, 150-165 (2014).
Kaliada, Dzianis, Götze, Friedrich, and Kukso, Olga. “The asymptotic number of integral cubic polynomials with bounded heights and discriminants”. Lithuanian Mathematical Journal 54.2 (2014): 150-165.