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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.

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The nonvanishing of certain character sums
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by S. Ullom PDF
Proc. Amer. Math. Soc. 45 (1974), 164-166 Request permission

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
  • Helmut Hasse, Über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952 (German). MR 0049239
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 45 (1974), 164-166
  • MSC: Primary 12A55; Secondary 10G05, 12A35
  • DOI: https://doi.org/10.1090/S0002-9939-1974-0354611-7
  • MathSciNet review: 0354611