Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Weighted short-interval character sums

Authors: Shigeru Kanemitsu, Hailong Li and Nianliang Wang
Journal: Proc. Amer. Math. Soc. 139 (2011), 1521-1532
MSC (2010): Primary 11L03, 11L26; Secondary 11B68, 11T24, 11S40
Published electronically: September 15, 2010
MathSciNet review: 2763742
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we shall establish the counterpart of Szmidt, Urbanowicz and Zagier's formula in the sense of the Hecker correspondence. The motivation is the derivation of the values of the Riemann zeta-function at positive even integral arguments from the partial fraction expansion for the hyperbolic cotangent function (or the cotangent function). Since the last is equivalent to the functional equation, we may view their elegant formula as one for the Lambert series, and comparing the Laurent coefficients, we may give a functional equational approach to the short-interval character sums with polynomial weight.

In view of the importance of these short-interval character sums, we assemble some handy formulations for them that are derived from Szmidt, Urbanowicz and Zagier's formula and Yamamoto's method, which also gives the conjugate sums. We shall also state the formula for the values of the Dirichlet $ L$-function with imprimitive characters.

References [Enhancements On Off] (What's this?)

  • 1. T. M. Apostol, Euler's $ \varphi$-function and separable Gauss sums, Proc. Amer. Math. Soc. 24 (1970), 482-485. MR 0257006 (41:1661)
  • 2. T. M. Apostol, Dirichlet $ L$-functions and Dirichlet characters, Proc. Amer. Math. Soc. 31 (1972), 384-386. MR 0285499 (44:2717)
  • 3. T. M. Apostol, Introduction to analytic number theory, Springer-Verlag, Berlin, 1976. MR 0434929 (55:7892)
  • 4. B. C. Berndt, Modular transformations and generalization of several formulas of Ramanujan, Rocky Mount. J. Math. 7 (1977), 147-189. MR 0429703 (55:2714)
  • 5. B. C. Berndt, Periodic Bernoulli numbers, summation formulas and applications, Theory and application of special functions, Academic Press, New York, 1975. MR 0389729 (52:10560)
  • 6. B. C. Berndt, Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (1975), 413-445. MR 0382187 (52:3075)
  • 7. B. C. Berndt, Classical theorems on quadratic residues, Enseign. Math. (2) 22 (1976), 261-304. MR 0441835 (56:229)
  • 8. B. C. Berndt and M. I. Knopp, Hecke's theory of modular forms and Dirichlet series, World Scientific, Singapore, 2008. MR 2387477 (2009a:11003)
  • 9. S. Bochner, Some properties of modular relations. Ann. of Math. (2) 53 (1951), 332-363. MR 0047719 (13:920b)
  • 10. P. E. Böhmer, Differenzengleichungen und Bestimmte Integrale, Koecher Verlag, Berlin, 1939.
  • 11. L. Carlitz, Arithmetic properties of generalized Bernoulli numbers, J. Reine Angew. Math. 202 (1959), 174-182. MR 0109132 (22:20)
  • 12. K. Chakraborty, S. Kanemitsu and H. Tsukada, Arithmetical Fourier series and the modular relation, to appear.
  • 13. P. Chowla, On the class-number of real quadratic fields, J. Reine Angew. Math. 230 (1968), 51-60. MR 0225752 (37:1345)
  • 14. K. Dilcher, L. Skula and I. Sh. Slavutsukii, Bernoulli Numbers Bibliography (1713-1990). Queen's University Press, Kingston, ON, 1991. MR 1119305 (92f:11001)
  • 15. H. Hasse, Vorlesungen über Zahlentheorie, 2 auf., Springer, Berlin-New York-Heidelberg, 1964. MR 0188128 (32:5569)
  • 16. T. Ishii and T. Oda, A short history on investigation of the special values of zeta and $ L$-functions of totally real number fields, in Proc. of the conf. in memory of Tsuneo Aarakawa, Automorphic forms and zeta functions, World Sci., Singapore, 2006. MR 2208776 (2007b:11180)
  • 17. K. Iwasawa, Lectures on $ p$-adic $ L$-functions, Ann. Math. Studies 74, Princeton Univ. Press, Princeton, 1972. MR 0360526 (50:12974)
  • 18. W. Johnson and K. J. Mitchell, Summation of sums of the Legendre symbols, Pacific J. Math. 69 (1977), 117-124. MR 0434936 (55:7899)
  • 19. H. Joris, On the evaluation of Gaussian sums for non-primitive Dirichlet characters, Enseign. Math. (2) 23 (1977), 13-18. MR 0441888 (56:279)
  • 20. S. Kanemitsu and Kuzumaki, Transformation formulas for Lambert series, Siaulai Math. Sem. 4 (2009), 105-123. MR 2530201
  • 21. S. Kanemitsu, J. Ma and Y. Tanigawa, Arithmetical identities and zeta-functions, Math. Nachr., to appear.
  • 22. S. Kanemitsu, J. Ma and W.-P. Zhang, On the discrete mean value of the product of two Dirichlet L-functions, Abh. Math. Sem. Univ. Hamburg 79 (2009), 149-164. MR 2545597
  • 23. S. Kanemitsu, Y. Tanigawa and M. Yoshimoto, Ramanujan's formula and modular forms, in Number-theoretic methods -- future trends (ed. by Shigeru Kanemitsu and Chaohua Jia), Kluwer Academic Publ., 2002, pp. 159-212. MR 1974140 (2004g:11036)
  • 24. S. Kanemitsu and H. Tsukada, Vistas of special functions, World Scientific, Singapore-London-New York, 2007. MR 2352572
  • 25. S. Kanemitsu, J. Urbanowicz and N. -L. Wang, Class number formula of a certain imaginary quadratic field, to appear.
  • 26. M. I. Knopp, Hamburger's theorem on $ \zeta(s)$ and the abundance principle for Dirichlet series with functional equations, Number Theory, ed. by R. P. Bambah et al., Hindustan Books Agency, 2000, 201-216. MR 1764804 (2001h:11112)
  • 27. D. S. Kubert and S. Lang, Modular units, Springer-Verlag, New York-Berlin, 1981. MR 648603 (84h:12009)
  • 28. H. W. Leopoldt, Eine Verallgemeinerung der Bernoullischen Zahlen, Abh. Math. Sem. Univ. Hamburg 22 (1958), 131-140. MR 0092812 (19:1161e)
  • 29. M. Lerch, Essais sur le calcul de nombre de classe de formes quadratiques binaires aux coefficients entiers, Acta Math. 29 (1905), 333-424; 30 (1906), 203-293.
  • 30. J. Milnor, On polylogarithms, Hurwitz zeta-functions and the Kubert identities, Enseign. Math. (2) 29 (1983), 281-322. MR 719313 (86d:11007)
  • 31. Y. Morita, On the Hurwitz-Lerch $ L$-function, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977), 29-43. MR 0441924 (56:315)
  • 32. J. Neukirch, The Beilinson conjecture for algebraic number fields, in ``Beilinson's conjectures on special values of $ L$-functions'', ed. by M. Rapoport et al., Academic Press, Boston, 1988, 193-247. MR 944995 (90f:11042)
  • 33. J.-P. Serre, A course in arithmetic, Springer-Verlag, Berlin-Heidelberg-New York, 1973. MR 0344216 (49:8956)
  • 34. A. Schinzel, J. Urbanowicz and P. van Wamelen, Class numbers and short sums of Kronecker symbols, J. Number Theory 78 (1999), 62-84. MR 1706925 (2000g:11103)
  • 35. J. Szmidt, J. Urbanowicz and D. Zagier, Congruences among generalized Bernoulli numbers, Acta Arith. 3 (1995), 273-278. MR 1339132 (96f:11032)
  • 36. H. S. Vandiver, An arithmetical theory of the Bernoulli numbers, Trans. Amer. Math. Soc. 51 (1942), 502-531. MR 0006742 (4:34e)
  • 37. N. -L. Wang, J. -Z. Li, D. -S. Liu, Euler number congruences and Dirichlet $ L$-functions, J. Number Theory 129 (2009), 1522-1531. MR 2521491 (2010c:11032)
  • 38. Y. Yamamoto, Dirichlet series with periodic coefficients, Proc. Intern. Sympos. ``Algebraic Number Theory'', Kyoto, 1976, 275-289. JSPS, Tokyo, 1977. MR 0450213 (56:8509)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 11L03, 11L26, 11B68, 11T24, 11S40

Retrieve articles in all journals with MSC (2010): 11L03, 11L26, 11B68, 11T24, 11S40

Additional Information

Shigeru Kanemitsu
Affiliation: Graduate School of Advanced Technology, Kinki University Iizuka, Fukuoka, Japan, 820-8555.

Hailong Li
Affiliation: Department of Mathematics, WeiNan Teachers College, WeiNan, People’s Republic of China, 714000.

Nianliang Wang
Affiliation: Institute of Mathematics, Shangluo University, Shangluo Shaanxi 726000, People’s Republic of China

Keywords: Dirichlet characters, Lambert series, Bernoulli numbers, short-interval character sums
Received by editor(s): October 28, 2009
Received by editor(s) in revised form: February 11, 2010, and May 4, 2010
Published electronically: September 15, 2010
Additional Notes: The authors were supported in part by JSPS grant No. 21540029 and by the NSF of Shaanxi Province (No. 2010JM1009).
Dedicated: Dedicated to Professor Masaaki Yoshida on his sixtieth birthday with great respect and friendship
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society