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Mathematics of Computation

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

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Real zeros of Dedekind zeta functions of real quadratic fields
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by Kok Seng Chua PDF
Math. Comp. 74 (2005), 1457-1470 Request permission

Abstract:

Let $\chi$ be a primitive, real and even Dirichlet character with conductor $q$, and let $s$ be a positive real number. An old result of H. Davenport is that the cycle sums $S_\nu (s,\chi )=\sum _{n=\nu q+1}^{(\nu +1)q-1} \frac {\chi (n)}{n^s}, \nu = 0,1,2,\dots ,$ are all positive at $s=1,$ and this has the immediate important consequence of the positivity of $L(1,\chi )$. We extend Davenport’s idea to show that in fact for $\nu \geq 1$, $S_\nu (s,\chi )>0$ for all $s$ with $1/2 \leq s \leq 1$ so that one can deduce the positivity of $L(s,\chi )$ by the nonnegativity of a finite sum $\sum _{\nu =0}^t S_\nu (s,\chi )$ for any $t \geq 0$. A simple algorithm then allows us to prove numerically that $L(s,\chi )$ has no positive real zero for a conductor $q$ up to 200,000, extending the previous record of 986 due to Rosser more than 50 years ago. We also derive various estimates explicit in $q$ of the $S_\nu (s,\chi )$ as well as the shifted cycle sums $T_\nu (s,\chi ):=\sum _{n=\nu q+\lfloor q/2 \rfloor +1}^{(\nu +1) q+\lfloor q/2 \rfloor } \frac {\chi (n)}{n^s}$ considered previously by Leu and Li for $s=1$. These explicit estimates are all rather tight and may have independent interests.
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Additional Information
  • Kok Seng Chua
  • Affiliation: Software and Computing Programme, Institute of High Performance Computing, 1 Science Park Road, #01-01, The Capricorn, Singapore Science Park II, Singapore 117528
  • Email: chuaks@ihpc.a-star.edu.sg
  • Received by editor(s): November 15, 2003
  • Received by editor(s) in revised form: February 21, 2004
  • Published electronically: July 21, 2004
  • © Copyright 2004 American Mathematical Society
  • Journal: Math. Comp. 74 (2005), 1457-1470
  • MSC (2000): Primary 11M20; Secondary 11M06
  • DOI: https://doi.org/10.1090/S0025-5718-04-01701-6
  • MathSciNet review: 2137012