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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

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Weighted short-interval character sums
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by Shigeru Kanemitsu, Hailong Li and Nianliang Wang PDF
Proc. Amer. Math. Soc. 139 (2011), 1521-1532 Request permission


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.

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Additional Information
  • Shigeru Kanemitsu
  • Affiliation: Graduate School of Advanced Technology, Kinki University Iizuka, Fukuoka, Japan, 820-8555.
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  • Hailong Li
  • Affiliation: Department of Mathematics, WeiNan Teachers College, WeiNan, People’s Republic of China, 714000.
  • Email:
  • Nianliang Wang
  • Affiliation: Institute of Mathematics, Shangluo University, Shangluo Shaanxi 726000, People’s Republic of China
  • Email:
  • 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
  • © Copyright 2010 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 139 (2011), 1521-1532
  • MSC (2010): Primary 11L03, 11L26; Secondary 11B68, 11T24, 11S40
  • DOI:
  • MathSciNet review: 2763742