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Real zeros of Dedekind zeta functions of real quadratic fields


Author: Kok Seng Chua
Journal: Math. Comp. 74 (2005), 1457-1470
MSC (2000): Primary 11M20; Secondary 11M06
DOI: https://doi.org/10.1090/S0025-5718-04-01701-6
Published electronically: July 21, 2004
MathSciNet review: 2137012
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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.


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

  • 1. P. Bateman and E. Grosswald, On Epstein's zeta function, Acta Arith., 9 (1964) 365-373.MR 31:3392
  • 2. J. B. Conrey and K. Soundararajan, Real zeros of quadratic Dirichlet L-functions, Invent. Math. 150 (2002) 1-44.MR 2004a:11089
  • 3. H. Davenport, On the series for $L(1)$, Journal of London Math. Soc., 24 (1949) 229-233.MR 11:162e
  • 4. H. Davenport, Multiplicative Number Theory, Second Edition Revised by H. Montgomery, Graduate Texts in Mathematics, 74, Springer-Verlag, New York-Berlin (1980).MR 2001f:11001
  • 5. M. G. Leu and W. Li, On the series for $L(1,\chi)$, Nagoya Math. J., Vol. 141 (1996), 125-142.MR 97c:11086
  • 6. M. Low, Real zeros of the Dedekind zeta functions of an imaginary quadratic field, Acta. Arith., 14 (1968), 117-140.MR 38:4425
  • 7. J. Rosser, Real zeros of real Dirichlet L-series, Bull. Amer. Math. Soc., 55, (1949) 906-913.MR 11:332c
  • 8. M. Rubinstein, http://www.ma.utexas.edu/users/miker/L_function/VALUES/
  • 9. P. Sarnak and A. Zaharescu, Some remarks on Landau-Siegel zeroes, Duke Math J., 111 (2002) no. 3, 495-507.MR 2002j:11097
  • 10. A. Selberg and S. Chowla, On Epstein's Zeta-function, J. Reine Angew. Math. 227 (1967), 86-110.MR 35:6632
  • 11. C. Siegel, Über die Classzahl quadratischer Zahlkörper, Acta. Arith. 1 (1935), 47-87.
  • 12. K. Soundararajan, Nonvanishing of quadratic Dirichlet L-function at $s=1/2$, Ann. of Math. 152 (2000) no. 2, 447-488.MR 2001k:11164
  • 13. M. Watkins, Real zeros of real odd Dirichlet $L$-functions, Math. Comp. 73 (2004), 415-423.

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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

DOI: https://doi.org/10.1090/S0025-5718-04-01701-6
Keywords: Real zeros of $L$ functions, Dirichlet series, primitive characters
Received by editor(s): November 15, 2003
Received by editor(s) in revised form: February 21, 2004
Published electronically: July 21, 2004
Article copyright: © Copyright 2004 American Mathematical Society

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