Abstract
Ramanujan sums have been studied and generalized by several authors. For example, Nowak [8] studied these sums over quadratic number fields, and Grytczuk [4] defined that on semigroups. In this note, we deduce some properties on sums of generalized Ramanujan sums and give examples on number fields. In particular, we have a relational expression between Ramanujan sums and residues of Dedekind zeta functions.
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T. M. Apostol, Arithmetical properties of generalized Ramanujan sums, Pacific J. Math., 41(2) (1972), 281–293.
T. H. Chan and A. V. Kumchev, On sums of Ramanujan sums, Acta arithm., 152 (2012), 1–10.
H. Cohen, Number theory. Vol. II. Analytic and modern tools, Graduate Texts in Mathematics, 240. Springer, New York, 2007.
A. Grytczuk, On Ramanujan sums on arithmetical semigroup, Tsukuba. J. Math., 16(2) (1992), 315–319.
I. Kiuchi and Y. Tanigawa, On arithmetic functions related to the Ramanujan sum, Period. Math. Hungar., 45(1–2) (2002), 87–99.
S. Lang, Algebraic number theory, 2nd edition, Graduate Texts in Mathematics, 110. Springer-Verlag, New York, 1994.
R. Murty and J. V. Order, Counting integral ideals in a number field, Expo. Math., 25 (2007), 53–66.
W. G. Nowak, The average size of Ramanujan sums over quadratic number fields, Arch. Math., 99 (2012), 433–442.
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd edition, revised by D. R. Heath-Brown, Oxford University Press, 1986.
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Fujisawa, Y. On sums of generalized Ramanujan sums. Indian J Pure Appl Math 46, 1–10 (2015). https://doi.org/10.1007/s13226-015-0103-1
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DOI: https://doi.org/10.1007/s13226-015-0103-1