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Proceedings of the American Mathematical Society
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
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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.


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Additional Information

Shigeru Kanemitsu
Affiliation: Graduate School of Advanced Technology, Kinki University Iizuka, Fukuoka, Japan, 820-8555.
Email: kanemitu@fuk.kindai.ac.jp

Hailong Li
Affiliation: Department of Mathematics, WeiNan Teachers College, WeiNan, People’s Republic of China, 714000.
Email: lihailong@wntc.edu.cn

Nianliang Wang
Affiliation: Institute of Mathematics, Shangluo University, Shangluo Shaanxi 726000, People’s Republic of China
Email: wangnianliangshangluo@yahoo.com.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10572-5
PII: S 0002-9939(2010)10572-5
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