Remote Access Mathematics of Computation
Green Open Access

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
DOI: https://doi.org/10.1090/S0025-5718-03-01537-0
Published electronically: May 7, 2003
MathSciNet review: 2034130
Full-text PDF Free Access

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?)

  • P. T. Bateman and E. Grosswald, On Epstein’s zeta function, Acta Arith. 9 (1964), 365–373. MR 179141, DOI https://doi.org/10.4064/aa-9-4-365-373
  • J. B. Conrey and K. Soundararajan, Real zeros of quadratic Dirichlet L-functions. Invent. Math. 150 (2002), 1–44.
  • M. E. Low, Real zeros of the Dedekind zeta function of an imaginary quadratic field, Acta Arith 14 (1967/1968), 117–140. MR 0236127, DOI https://doi.org/10.4064/aa-14-2-117-140
  • G. Purdy, The real zeros of the Epstein zeta function. Ph. D. thesis. Univ. of Illinois (1972).
  • G. Rieger, Review of [M. Low, Real zeros of the Dedekind zeta function of an imaginary quadratic field. Acta Arith. 14 (1968), 117-140], Math. Reviews 38/4425 (1970).

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 11M20, 11M06

Retrieve articles in all journals with MSC (2000): 11M20, 11M06


Additional Information

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

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