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 | References | Similar Articles | Additional Information

Abstract: Let be a primitive, real and even Dirichlet character with conductor , and let be a positive real number. An old result of H. Davenport is that the cycle sums are all positive at and this has the immediate important consequence of the positivity of . We extend Davenport's idea to show that in fact for , for all with so that one can deduce the positivity of by the nonnegativity of a finite sum for any . A simple algorithm then allows us to prove numerically that has no positive real zero for a conductor 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 of the as well as the shifted cycle sums considered previously by Leu and Li for . These explicit estimates are all rather tight and may have independent interests.

**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 , 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 , 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 , Ann. of Math. 152 (2000) no. 2, 447-488.MR**2001k:11164****13.**M. Watkins, Real zeros of real odd Dirichlet -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