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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Explicit upper bounds for $ L$-functions on the critical line


Author: Vorrapan Chandee
Journal: Proc. Amer. Math. Soc. 137 (2009), 4049-4063
MSC (2000): Primary 11M41; Secondary 11E25
Published electronically: August 7, 2009
MathSciNet review: 2538566
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Abstract: We find an explicit upper bound for general $ L$-functions on the critical line, assuming the Generalized Riemann Hypothesis, and give as illustrative examples its application to some families of $ L$-functions and Dedekind zeta functions. Further, this upper bound is used to obtain lower bounds beyond which all eligible integers are represented by Ramanujan's ternary form and Kaplansky's ternary forms. This improves on previous work by Ono and Soundararajan on Ramanujan's form and by Reinke on Kaplansky's forms with a substantially easier proof.


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Additional Information

Vorrapan Chandee
Affiliation: Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, California 94305
Email: vchandee@math.stanford.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10075-8
PII: S 0002-9939(09)10075-8
Keywords: $L$-functions, critical line, ternary quadratic form
Received by editor(s): April 15, 2009
Received by editor(s) in revised form: April 20, 2009
Published electronically: August 7, 2009
Communicated by: Ken Ono
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.