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ISSN 1088-6826(e) ISSN 0002-9939(p)

Explicit upper bounds for -functions on the critical line

Author(s): Vorrapan Chandee
Journal: Proc. Amer. Math. Soc. 137 (2009), 4049-4063.
MSC (2000): Primary 11M41; Secondary 11E25
Posted: August 7, 2009
MathSciNet review: 2538566
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Abstract: We find an explicit upper bound for general -functions on the critical line, assuming the Generalized Riemann Hypothesis, and give as illustrative examples its application to some families of -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: 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
Posted: August 7, 2009
Communicated by: Ken Ono
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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