Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The nonvanishing of certain character sums


Author: S. Ullom
Journal: Proc. Amer. Math. Soc. 45 (1974), 164-166
MSC: Primary 12A55; Secondary 10G05, 12A35
MathSciNet review: 0354611
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \chi $ be a Dirichlet character with conductor $ f$ and $ M(\chi ) = \Sigma a\bar \chi (a)$, summation over integers $ a$ prime to $ f$ and $ 1 \leqslant a < f$. It is well known that the nonvanishing of the Dirichlet $ L$-function $ L(s,\chi )$ at $ s = 1$ implies $ M(\chi ) \ne 0$ for $ \chi $ imaginary, i.e. $ \chi ( - 1) = - 1$. This article provides a purely algebraic proof that $ M(\chi ) \ne 0$ when the conductor $ f$ is a prime power and the imaginary $ \chi $ is either a faithful character or has order a power of 2.


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


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 12A55, 10G05, 12A35

Retrieve articles in all journals with MSC: 12A55, 10G05, 12A35


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0354611-7
PII: S 0002-9939(1974)0354611-7
Keywords: Character sum, cyclotomic field, class number, integral group ring
Article copyright: © Copyright 1974 American Mathematical Society