Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Weighted short-interval character sums
HTML articles powered by AMS MathViewer

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

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
Similar Articles
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
  • 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: https://doi.org/10.1090/S0002-9939-2010-10572-5
  • MathSciNet review: 2763742