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Class Number Formula for Certain Imaginary Quadratic Fields

Received: 26 October 2014     Accepted: 6 November 2014     Published: 29 November 2014
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Abstract

In this note we shall show how Carlitz in 1954 could have reached an analogue of the Voronoi congruence in the more difficult case of p≡1(mod4): h(-4p) ≡B(p+1)/2(x4)(mod p), where B(p+1)/2(x4) is the generalized Bernoulli number with x4 being the Kronecker symbol associated to the Gaussian field Q(√-4).

Published in Pure and Applied Mathematics Journal (Volume 4, Issue 2-1)

This article belongs to the Special Issue Abridging over Troubled Water---Scientific Foundation of Engineering Subjects

DOI 10.11648/j.pamj.s.2015040201.11
Page(s) 1-6
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2014. Published by Science Publishing Group

Keywords

Class Number Formula, Short Interval Character Sum, Generalized Bernoulli Number, Euler Number

References
[1] Z. N. Borevic and I. S. Shafarevic, The theory of numbers, 2nd edition, Nauka, Moscow 1972 (in Russian); first edition published in 1964.
[2] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York etc., 1990.
[3] R. K. Guy, Unsolved Probelms in Number Theory, 3rd ed. Springer-Verlag, New York etc., 2004.
[4] G. -D. Liu, Generating functions and generalized Euler numbers, doctoral thesis, Kinki University, 2006.
[5] N. -L. Wang, J. -Z. Li and D. -S. Liu, Euler numbers congruences and Dirichlet L-functions, J. Number Theory 129 (2009), 1522-1531.
[6] A. Schinzel, J. Urbanowicz and P. Van Wamelen, Class numbers and short sums of Kronecker symbols, J. Number Theory 78 (1999), 62-84.
[7] J. Szmidt, J. Urbanowicz and D. Zagier, Congruences among generalized Bernoulli numbers, Acta Arith. 71 (1995), 275-278.
[8] W. -P. Zhang and Z.-F. Xu, On a conjecture of the Euler numbers, J. Number Theory 127 (2007), 283-291.
[9] T, Funakura, On Kronecker's limit formula for Dirichlet series with periodic coefficients, Acta Arith. 55 (1990), 59-73.
[10] K. Chakraborty, S. Kanemitsu and T. Kuzumaki, Arithmetical class number formula for certain quadratic fields, Hardy-Ramanujan J. 36 (2013), 1-7
[11] H. Davenport, Multiplicative Number Theory, Markham 1967, second edition Springer-Verlag, New York etc., 1982.
[12] B. C. Berndt, Classical theorems on quadratic residues, Enseign. Math. (2) 22(1976), 261-304.
[13] L. E. Dickson, History of Number Theory, Chelsea, New York 1952.
[14] P. Chowla, On the class number of real quadratic field, J. Reine Angew. Math. 230 (1968), 51-60.
[15] L. Carlitz, A note on Euler numbers and polynomials, Nagoya Math. J. 7 (1954), 35-43.
[16] Y. Yamamoto, Dirichlet series with periodic coefficients, Proc. Intern. Sympos. ``Algebraic Number Theory", Kyoto 1976, JSPS, Tokyo 1977, 275-289.
[17] B. C. Berndt, Periodic Bernoulli numbers, summation formulas and applications, Theory and Applications of Special Functions, Richard Askey ed. Academic Press, New York, 1975, 143-189.
[18] L. Carlitz, Arithmetical properties of generalized Bernoulli numbers, J. Reine Angew. Math. 202 (1959), 174-182.
[19] N. Nielsen, Traite elementaire des Nombres de Bernoulli, Gauther-Villars, Paris, 1923, 35-43.
[20] B. C. Berndt, Character analogues of the Poisson and Euler-Maclaurin summation formulas with applications, J. Number Theory 7(1975), 413-445.
[21] K. Shiratani and S. Yokoyama, An application of p-adic convolutions, Mem. Fac. Sci., Kyushu Univ. Ser. Math. 36 (1982), 73-83.
[22] L. C. Washington, Introduction to cyclotomic fields, 2nd ed., Springer, New York/Berlin, 1997.
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  • APA Style

    N. L. Wang, T. Arai. (2014). Class Number Formula for Certain Imaginary Quadratic Fields. Pure and Applied Mathematics Journal, 4(2-1), 1-6. https://doi.org/10.11648/j.pamj.s.2015040201.11

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    ACS Style

    N. L. Wang; T. Arai. Class Number Formula for Certain Imaginary Quadratic Fields. Pure Appl. Math. J. 2014, 4(2-1), 1-6. doi: 10.11648/j.pamj.s.2015040201.11

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    AMA Style

    N. L. Wang, T. Arai. Class Number Formula for Certain Imaginary Quadratic Fields. Pure Appl Math J. 2014;4(2-1):1-6. doi: 10.11648/j.pamj.s.2015040201.11

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  • @article{10.11648/j.pamj.s.2015040201.11,
      author = {N. L. Wang and T. Arai},
      title = {Class Number Formula for Certain Imaginary Quadratic Fields},
      journal = {Pure and Applied Mathematics Journal},
      volume = {4},
      number = {2-1},
      pages = {1-6},
      doi = {10.11648/j.pamj.s.2015040201.11},
      url = {https://doi.org/10.11648/j.pamj.s.2015040201.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.s.2015040201.11},
      abstract = {In this note we shall show how Carlitz in 1954 could have reached an analogue of the Voronoi congruence in the more difficult case of p≡1(mod4): h(-4p) ≡B(p+1)/2(x4)(mod p), where B(p+1)/2(x4) is the generalized Bernoulli number with x4 being the Kronecker symbol associated to the Gaussian field Q(√-4).},
     year = {2014}
    }
    

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    AB  - In this note we shall show how Carlitz in 1954 could have reached an analogue of the Voronoi congruence in the more difficult case of p≡1(mod4): h(-4p) ≡B(p+1)/2(x4)(mod p), where B(p+1)/2(x4) is the generalized Bernoulli number with x4 being the Kronecker symbol associated to the Gaussian field Q(√-4).
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Author Information
  • Dept. of Appl. Math., Shangluo Univ. Shangluo,726000, PRC

  • Dept. of Appl. Math., Shangluo Univ. Shangluo,726000, PRC

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