Abstract
LetK be a cubic number field. Denote byA K (x) the number of ideals with ideal norm ≤x, and byQ K (x) the corresponding number of squarefree ideals. The following asymptotics are proved. For every ε>0 ε>0
Herec 1,c 2 andc are positive constants. Assuming the Riemann hypotheses for the Dedekind zeta function ζ K , the error term in the second result can be improved toO(x 53/116+ε).
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Atkinson, F. V.: A divisor problem. Quart. J. Oxford12, 193–200 (1941).
Hasse, H.: Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper. Teil I. Jber. Dtsch. Math. Ver.XXXV, 1–55 (1926).
Hasse, H.: Arithmetische Theorie der kubischen Zahlkörper auf klassenkörpertheoretischer Grundlage. Math. Z.31, 565–582 (1929).
Kolesnik, G.: On the estimation of multiple exponential sums. In: Recent Progress in Analytic Number Theory, Vol. I, pp. 231–246 (Halberstam, H., Holley, L.; eds.). London: Academic Press. 1981.
Landau, E.: Einführung in die elementare und analytische Theorie der algebraischen Zahlen und der Ideale. New York: Chelsea. 1949.
Mitsui, T.: On the prime ideal theorem. J. Math. Soc. Japan20, 233–247 (1968).
Nowak, W. G.: Zur Verteilung der quadratfreien Ideale in quadratischen Zahlkörpern. Abh. Braunschw. wiss. Ges.39, 31–36 (1987).
Nowak, W. G., Schmeier, M.: Conditional asymptotic formulae for a class of arithmetic functions. Proc. Amer. Math. Soc.103 (To appear.)
Prachar, K.: Primzahlverteilung. Berlin-Heidelberg-New York: Springer. 1957.
Rademacher, H.: On the Phragmen-Lindelöf theorem and some applications. Math. Z.72, 192–204 (1959).
Walfisz, A.: Weylsche Exponentialsummen in der neueren Zahlentheorie. Berlin: VEB. 1962.
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Müller, W. On the distribution of ideals in cubic number fields. Monatshefte für Mathematik 106, 211–219 (1988). https://doi.org/10.1007/BF01318682
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DOI: https://doi.org/10.1007/BF01318682