Abstract
In this paper, we consider Ramanujan’s sums over arbitrary Dedekind domain with finite norm property. We define the Ramanujan’s sums η(a, A) and η(B, A), where a is an arbitrary element in a Dedekind domain, B is an ideal and A is a non-zero ideal. In particular, we discuss the Kluyver formula and Hölder formula for η(a, A) and η(B, A). We also prove the reciprocity formula enjoyed by η(B, A) and the orthogonality relations for η(a, A) in the last two parts.
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References
Apostol, T. M.: Introduction to Analytic Number Theory, Springer-Verlag, New York, 1976
Cohen, E.: An extension of Ramanujan’s sum. Duke Math. J., 16, 85–90 (1949)
Cohen, E.: An extension of Ramanujan’s sum II, Additive properties. Duke Math. J., 22, 543–550 (1955)
Cohen, E.: An extension of Ramanujan’s sum III, Connection with totient functions. Duke Math. J., 23, 623–630 (1956)
Diaconis, P., Isaacs, I. M.: Supercharacters and super-classes for algebra groups. Trans. Amer. Math. Soc., 360, 2359–2392 (2008)
Fowler, C., Garcia, S., Karaali, G.: Ramanujan sums as super super-characters. Ramanujan J., 35, 205–241 (2014)
Grytczuk, A.: On Ramanujan Sums on arithmetical semigroups. Tsukuba J. Math., 16, 315–319 (1992)
Johnson, R. K.: A reciprocity law for Ramanujan sums. Pacific J. Math., 98, 99–105 (1982)
Johnson, R. K.: Reciprocity in Ramanujan sums. Math. Mag., 59, 216–222 (1986)
Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers, Springer-Verlag, Berlin, 2004
Nowak, W. G.: The average size of Ramanujan sums over quadratic number fields. Arch. Math., 99, 435–442 (2012)
Ramaiah, V. S.: Arithmetical sums in regular convolutions. J. Reine Angew. Math., 303–304, 265–283 (1978)
Wang, Y. J., Ji, C. G.: Ramanujan’s sum in the ring of integer of an algebraic number field. Int. J. Number Theory, 16, 65–76 (2020)
Zariski, O., Samuel, P.: Commutative Algebra, Vol. 1, Springer-Verlag, New York, 1965
Zheng, Z. Y.: On the polynomial Ramanujan sums over finite fields. Ramanujan J., 46, 863–898 (2018)
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We thank the referees for their time and comments.
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Supported by the National Research and Development Program of China (Grant No. 2018YFB1107402)
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Zheng, Z.Y., Chen, M. & Hong, Z.W. On Ramanujan Sums over a Dedekind Domain with Finite Norm Property. Acta. Math. Sin.-English Ser. 39, 149–160 (2023). https://doi.org/10.1007/s10114-023-2156-0
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DOI: https://doi.org/10.1007/s10114-023-2156-0