Abstract
In 2008, Deza and Varukhina established asymptotic formula for the mean value of the arithmetic function \(\tau _{k_1}^K(n)\tau _{k_2}^K(n)\cdots \tau _{k_l}^K(n)\), where \(K\) is a quadratic or cyclotomic field, and \(\tau _{k}^K(n)\) is the \(k\)-dimensional divisor function in the number field \(K\). Recently, Lü generalized their results to any Galois extension \(K\) of the rational field. It seems interesting to deal with similar problems which involve different number fields. In this paper, we are concerned with the mean value of the arithmetic function \(\tau _{k_1}^{K_1}(n)\tau _{k_2}^{K_2}(n)\cdots \tau _{k_l}^{K_l}(n)\), where \(K_j\) are number fields whose discriminants are relatively prime.
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References
Chandraseknaran, K., Good, A.: On the number of integral ideals in Galois extensions. Monatsh. Math. 95, 99–109 (1983)
Chandraseknaran, K., Narasimhan, R.: The approximate functional equation for a class of zeta-functions. Math. Ann. 152, 30–64 (1963)
Deza, E., Varukhina, L.: On mean values of some arithmetic functions in number fields. Discrete. Math. 308, 4892–4899 (2008)
Heath-Brown, D.R.: The growth rate of the Dedekind zeta-function on the critical line. Acta Arith. 49, 323–339 (1988)
Huxley, M.N., Watt, N.: The number of ideals in a quadratic field II. Israel J. Math. Part A 120, 125–153 (2000)
Ivić, A.: The Riemann Zeta-Function. Wiley, New York (1985)
Iwaniec, H., Kowalski, E.: Analytic Number Theory, Amer. Math. Soc. Colloquium Publ., vol. 53. Amer. Math. Soc., Providence (2004)
Landau, E.: Einführung in die elementare umd analytische Theorie der algebraischen Zahlen und der Ideals. New York (1949)
Lü, G.S.: On mean values of some arithmetic functions in number fields. Acta Math. Hung. 132, 1924–1938 (2011)
Müller, W.: On the distribution of ideals in cubic number fields. Monatsh. Math. 106, 211–219 (1988)
Nowak, W.G.: On the distribution of integral ideals in algebraic number theory fields. Math. Nachr. 161, 59–74 (1993)
Panteleeva, E.: Dirichlet divisor problem in number fields. Math. Notes 3(44), 750–757 (1988)
Panteleeva, E.: On the mean values of certain arithmetic functions. Math. Notes 2(55), 178–184 (1994)
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This work is supported in part by the National Natural Science Foundation of China (11031004, 11171182), NCET (NCET-10-0548) and Shandong Province Natural Science Foundation for Distinguished Young Scholars (JQ201102).
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Lü, G., Ma, W. On mean values of some arithmetic functions involving different number fields. Ramanujan J 38, 101–113 (2015). https://doi.org/10.1007/s11139-014-9612-5
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DOI: https://doi.org/10.1007/s11139-014-9612-5