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 |
Class Number Formula, Short Interval Character Sum, Generalized Bernoulli Number, Euler Number
| [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. |
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
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
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
Share This Article
Twitter
Linked In
Facebook
Copy
Download@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}
}
TY - JOUR T1 - Class Number Formula for Certain Imaginary Quadratic Fields AU - N. L. Wang AU - T. Arai Y1 - 2014/11/29 PY - 2014 N1 - https://doi.org/10.11648/j.pamj.s.2015040201.11 DO - 10.11648/j.pamj.s.2015040201.11 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 1 EP - 6 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.s.2015040201.11 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). VL - 4 IS - 2-1 ER -
Email: