Explicit upper bounds for -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 | References | Similar Articles | Additional Information

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:
https://doi.org/10.1090/S0002-9939-09-10075-8

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.