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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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  • 1. L. Ahlfors, Complex Analysis, McGraw-Hill, New York, 1978. MR 510197 (80c:30001)
  • 2. H. Davenport, Multiplicative Number Theory, Graduate Texts in Mathematics, vol. 74, Springer-Verlag, New York, 2000. MR 1790423 (2001f:11001)
  • 3. W. Duke and R. Schulze-Pillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99, 1, 49-57, 1990. MR 1029390 (90m:11051)
  • 4. G. Harcos, Uniform approximate functional equation for principal $ L$-functions, Internat. Math. Res. Notices 18, 923-932, 2002. MR 1902296 (2003d:11074)
  • 5. H. Iwaniec and E. Kowalski, Analytic Number Theory, American Mathematical Society Colloquium Publications, vol. 53, Amer. Math. Soc., Providence, RI, 2004. MR 2061214 (2005h:11005)
  • 6. J. Kelley, Kaplansky's ternary quadratic form, Int. J. Math. Math. Sci. 25, 289-292, 2001. MR 1812392 (2002c:11038)
  • 7. K. Ono and K. Soundararajan, Ramanujan's ternary quadratic form. Invent. Math. 130, no. 3, 415-454, 1997. MR 1483991 (99b:11036)
  • 8. T. Reinke, Darstellbarkeit ganzer Zahlen durch Kaplanskys tern $ \it {\ddot{a}}$re quadratische Form, Ph.D. thesis, Fachbereich Mathematik und Informatik, Universität Münster, 2003.
  • 9. J. Barkley Rosser and L. Schoenfeld, Sharper bounds for the Chebyshev functions $ \theta(x)$ and $ \psi(x)$, Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29, 243-269, 1975. MR 0457373 (56:15581a)
  • 10. G. Shimura, On modular forms of half integral weight. Ann. of Math. (2) 97, 440-481, 1973. MR 0332663 (48:10989)
  • 11. K. Soundararajan, Moments of the Riemann zeta-function, Ann. of Math. (2) 170, 2009.
  • 12. H. M. Stark, The analytic theory of algebraic numbers. Bull. Amer. Math. Soc. 81, no. 6, 961-972, 1975. MR 0444611 (56:2961)
  • 13. E. C. Titchmarsh, The Theory of the Riemann Zeta-function, Oxford University Press, New York, 1986. MR 882550 (88c:11049)

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Additional Information

Vorrapan Chandee
Affiliation: Department of Mathematics, Stanford University, 450 Serra Mall, Building 380, Stanford, California 94305

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.

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