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On mean values of some arithmetic functions involving different number fields

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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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Correspondence to Guangshi Lü.

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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

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