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Mean values and moments of arithmetic functions over number fields

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For an odd integer \(d > 1\) and a finite Galois extension \(K/{\mathbb {Q}}\) of degree d, Lü and Yang (J Number Theory 131:1924–1938, 2011) obtained an asymptotic formula for the mean values of the divisor function for K over square integers. In this article, we obtain the same for finitely many number fields of odd degree and pairwise coprime discriminants, together with the moment of the error term arising here, following the method adapted by Shi (An Stiint Univ Al I Cuza Iasi Mat (N.S.) 62:615–621, 2016). We also define the sum of divisor function over number fields and find the asymptotic behaviour of the summatory function of two number fields taken together.

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Acknowledgements

We are grateful to Prof. R. Thangadurai and Dr. Anirban Mukhopadhyay for their support, encouragement throughout the project and going through the manuscript thoroughly. The second author sincerely acknowledges Prof. R. Balasubramanian and Mr. Priyamvad Srivastav for having several fruitful discussions. The work has been completed when the second author was visiting Harish-Chandra Research Institute and he acknowledges the excellent hospitality and facilities provided by the Institute.

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  1. Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Allahabad, 211019, India

    Jaitra Chattopadhyay

  2. Institute of Mathematical Sciences, HBNI, CIT Campus, Taramani, Chennai, 600113, India

    Pranendu Darbar

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  1. Jaitra Chattopadhyay
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  2. Pranendu Darbar
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Correspondence to Jaitra Chattopadhyay.

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Chattopadhyay, J., Darbar, P. Mean values and moments of arithmetic functions over number fields. Res. number theory 5, 23 (2019). https://doi.org/10.1007/s40993-019-0160-3

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