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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Real zeros of real odd Dirichlet $L$-functions


Author: Mark Watkins
Journal: Math. Comp. 73 (2004), 415-423
MSC (2000): Primary 11M20; Secondary 11M06
Published electronically: May 7, 2003
MathSciNet review: 2034130
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $\chi$ be a real odd Dirichlet character of modulus $d$, and let $L(s,\chi)$ be the associated Dirichlet $L$-function. As a consequence of the work of Low and Purdy, it is known that if $d\le 800\,000$ and $d\neq 115\,147$, $357\,819$, $636\,184$, then $L(s,\chi)$ has no positive real zeros. By a simple extension of their ideas and the advantage of thirty years of advances in computational power, we are able to prove that if $d\le 300\,000\,000$, then $L(s,\chi)$ has no positive real zeros.


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

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  • 2. J. B. Conrey and K. Soundararajan, Real zeros of quadratic Dirichlet L-functions. Invent. Math. 150 (2002), 1-44.
  • 3. M. E. Low, Real zeros of the Dedekind zeta function of an imaginary quadratic field, Acta Arith 14 (1967/1968), 117–140. MR 0236127 (38 #4425)
  • 4. G. Purdy, The real zeros of the Epstein zeta function. Ph. D. thesis. Univ. of Illinois (1972).
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Additional Information

Mark Watkins
Affiliation: Department of Mathematics, McAllister Building, The Pennsylvania State University, University Park, Pennsylvania 16802
Email: watkins@math.psu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01537-0
PII: S 0025-5718(03)01537-0
Received by editor(s): February 14, 2002
Received by editor(s) in revised form: May 29, 2002
Published electronically: May 7, 2003
Article copyright: © Copyright 2003 American Mathematical Society