DOI QR코드

DOI QR Code

EQUIDISTRIBUTION OF HIGHER DIMENSIONAL GENERALIZED DEDEKIND SUMS AND EXPONENTIAL SUMS

  • Chae, Hi-joon (Department of Mathematics Education Hongik University) ;
  • Jun, Byungheup (Department of Mathematical Sciences Ulsan National Institute of Science and Technology) ;
  • Lee, Jungyun (Department of Mathematics Education Kangwon National University)
  • 투고 : 2019.06.11
  • 심사 : 2019.09.11
  • 발행 : 2020.07.01

초록

We consider generalized Dedekind sums in dimension n, defined as sum of products of values of periodic Bernoulli functions. For the generalized Dedekind sums, we associate a Laurent polynomial. Using this, we associate an exponential sum of a Laurent polynomial to the generalized Dedekind sums and show that this exponential sum has a nontrivial bound that is sufficient to fulfill the equidistribution criterion of Weyl and thus the fractional part of the generalized Dedekind sums are equidistributed in ℝ/ℤ.

키워드

과제정보

The first author was supported by 2019 Hongik University Research Fund. The second author was supported by NRF-2018R1D1A1A02085748. The third author was supported by 2019 Research Grant from Kangwon National University and NRF-2017R1A6A3A11030486.

참고문헌

  1. T. M. Apostol, Generalized Dedekind sums and transformation formulae of certain Lambert series, Duke Math. J. 17 (1950), 147-157. http://projecteuclid.org/euclid.dmj/1077476005 https://doi.org/10.1215/S0012-7094-50-01716-9
  2. M. Brion and M. Vergne, Lattice points in simple polytopes, J. Amer. Math. Soc. 10 (1997), no. 2, 371-392. https://doi.org/10.1090/S0894-0347-97-00229-4
  3. L. Carlitz, Some theorems on generalized Dedekind sums, Pacific J. Math. 3 (1953), 513-522. http://projecteuclid.org/euclid.pjm/1103051325 https://doi.org/10.2140/pjm.1953.3.513
  4. R. Chapman, Reciprocity laws for generalized higher dimensional Dedekind sums, Acta Arith. 93 (2000), no. 2, 189-199. https://doi.org/10.4064/aa-93-2-189-199
  5. R. Dedekind, Erlauterungen zu zwei Fragmenten von Riemann, Riemann's Gesammelte Mathematische Werke, 2nd edition, 1892.
  6. P. Deligne, Applications de la formule des traces aux sommes trigonometriques, in Cohomologieetale, 168-232, Lecture Notes in Math., 569, Springer, Berlin, 1977.
  7. J. Denef and F. Loeser, Weights of exponential sums, intersection cohomology, and Newton polyhedra, Invent. Math. 106 (1991), no. 2, 275-294. https://doi.org/10.1007/BF01243914
  8. S. Garoufalidis and J. E. Pommersheim, Values of zeta functions at negative integers, Dedekind sums and toric geometry, J. Amer. Math. Soc. 14 (2001), no. 1, 1-23. https://doi.org/10.1090/S0894-0347-00-00352-0
  9. G. van der Geer, Hilbert Modular Surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 16, Springer-Verlag, Berlin, 1988. https://doi.org/10.1007/978-3-642-61553-5
  10. D. Hickerson, Continued fractions and density results for Dedekind sums, J. Reine Angew. Math. 290 (1977), 113-116. https://doi.org/10.1515/crll.1977.290.113
  11. S. Hu and D. Solomon, Properties of higher-dimensional Shintani generating functions and cocycles on $PGL_3(Q)$, Proc. London Math. Soc. (3) 82 (2001), no. 1, 64-88. https://doi.org/10.1112/S0024611500012612
  12. H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, 53, American Mathematical Society, Providence, RI, 2004. https://doi.org/10.1090/coll/053
  13. B. Jun and J. Lee, Equidistribution of generalized Dedekind sums and exponential sums, J. Number Theory 137 (2014), 67-92. https://doi.org/10.1016/j.jnt.2013.10.020
  14. B. Jun and J. Lee, Special values of partial zeta functions of real quadratic fields at nonpositive integers and the Euler-Maclaurin formula, Trans. Amer. Math. Soc. 368 (2016), no. 11, 7935-7964. https://doi.org/10.1090/tran/6679
  15. C. Meyer, Die Berechnung der Klassenzahl Abelscher Korper uber quadratischen Zahlkorpern, Akademie-Verlag, Berlin, 1957.
  16. G. Myerson, Dedekind sums and uniform distribution, J. Number Theory 28 (1988), no. 3, 233-239. https://doi.org/10.1016/0022-314X(88)90039-X
  17. A. N. Parsin, On the arithmetic of two-dimensional schemes. I. Distributions and residues, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 4, 736-773, 949.
  18. J. E. Pommersheim, Barvinok's algorithm and the Todd class of a toric variety, J. Pure Appl. Algebra 117/118 (1997), 519-533. https://doi.org/10.1016/S0022-4049(97)00025-X
  19. H. Rademacher and E. Grosswald, Dedekind Sums, The Mathematical Association of America, Washington, DC, 1972.
  20. C. L. Siegel, Bernoullische Polynome und quadratische Zahlkorper, Nachr. Akad. Wiss. Gottingen Math.-Phys. Kl. II (1968), 7-38.
  21. I. Vardi, A relation between Dedekind sums and Kloosterman sums, Duke Math. J. 55 (1987), no. 1, 189-197. https://doi.org/10.1215/S0012-7094-87-05510-4
  22. H. Weyl, Uber die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313-352. https://doi.org/10.1007/BF01475864
  23. D. Zagier, Higher dimensional Dedekind sums, Math. Ann. 202 (1973), 149-172. https://doi.org/10.1007/BF01351173
  24. D. Zagier, Valeurs des fonctions zeta des corps quadratiques reels aux entiers negatifs, in Journees Arithmetiques de Caen (Univ. Caen, Caen, 1976), 135-151. Asterisque 41-42, Soc. Math. France, Paris, 1977.