Explicit upper bounds for functions on the critical line
Author:
Vorrapan Chandee
Journal:
Proc. Amer. Math. Soc. 137 (2009), 40494063
MSC (2000):
Primary 11M41; Secondary 11E25
Published electronically:
August 7, 2009
MathSciNet review:
2538566
Fulltext PDF Free Access
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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.
 1.
Lars
V. Ahlfors, Complex analysis, 3rd ed., McGrawHill Book Co.,
New York, 1978. An introduction to the theory of analytic functions of one
complex variable; International Series in Pure and Applied Mathematics. MR 510197
(80c:30001)
 2.
Harold
Davenport, Multiplicative number theory, 3rd ed., Graduate
Texts in Mathematics, vol. 74, SpringerVerlag, New York, 2000.
Revised and with a preface by Hugh L. Montgomery. MR 1790423
(2001f:11001)
 3.
William
Duke and Rainer
SchulzePillot, Representation of integers by positive ternary
quadratic forms and equidistribution of lattice points on ellipsoids,
Invent. Math. 99 (1990), no. 1, 49–57. MR 1029390
(90m:11051), http://dx.doi.org/10.1007/BF01234411
 4.
Gergely
Harcos, Uniform approximate functional equation for principal
𝐿functions, Int. Math. Res. Not. 18 (2002),
923–932. MR 1902296
(2003d:11074), http://dx.doi.org/10.1155/S1073792802111184
 5.
Henryk
Iwaniec and Emmanuel
Kowalski, Analytic number theory, American Mathematical
Society Colloquium Publications, vol. 53, American Mathematical
Society, Providence, RI, 2004. MR 2061214
(2005h:11005)
 6.
James
Kelley, Kaplansky’s ternary quadratic form, Int. J.
Math. Math. Sci. 25 (2001), no. 5, 289–292. MR 1812392
(2002c:11038), http://dx.doi.org/10.1155/S0161171201005294
 7.
Ken
Ono and K.
Soundararajan, Ramanujan’s ternary quadratic form,
Invent. Math. 130 (1997), no. 3, 415–454. MR 1483991
(99b:11036), http://dx.doi.org/10.1007/s002220050191
 8.
T. Reinke, Darstellbarkeit ganzer Zahlen durch Kaplanskys tern re quadratische Form, Ph.D. thesis, Fachbereich Mathematik und Informatik, Universität Münster, 2003.
 9.
J.
Barkley Rosser and Lowell
Schoenfeld, Sharper bounds for the Chebyshev
functions 𝜃(𝑥) and 𝜓(𝑥), Math. Comp. 29 (1975), 243–269.
Collection of articles dedicated to Derrick Henry Lehmer on the occasion of
his seventieth birthday. MR 0457373
(56 #15581a), http://dx.doi.org/10.1090/S00255718197504573737
 10.
Goro
Shimura, On modular forms of half integral weight, Ann. of
Math. (2) 97 (1973), 440–481. MR 0332663
(48 #10989)
 11.
K. Soundararajan, Moments of the Riemann zetafunction, Ann. of Math. (2) 170, 2009.
 12.
H.
M. Stark, The analytic theory of algebraic
numbers, Bull. Amer. Math. Soc.
81 (1975), no. 6,
961–972. MR 0444611
(56 #2961), http://dx.doi.org/10.1090/S000299041975138730
 13.
E.
C. Titchmarsh, The theory of the Riemann zetafunction, 2nd
ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited
and with a preface by D. R. HeathBrown. MR 882550
(88c:11049)
 1.
 L. Ahlfors, Complex Analysis, McGrawHill, New York, 1978. MR 510197 (80c:30001)
 2.
 H. Davenport, Multiplicative Number Theory, Graduate Texts in Mathematics, vol. 74, SpringerVerlag, New York, 2000. MR 1790423 (2001f:11001)
 3.
 W. Duke and R. SchulzePillot, Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Invent. Math. 99, 1, 4957, 1990. MR 1029390 (90m:11051)
 4.
 G. Harcos, Uniform approximate functional equation for principal functions, Internat. Math. Res. Notices 18, 923932, 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, 289292, 2001. MR 1812392 (2002c:11038)
 7.
 K. Ono and K. Soundararajan, Ramanujan's ternary quadratic form. Invent. Math. 130, no. 3, 415454, 1997. MR 1483991 (99b:11036)
 8.
 T. Reinke, Darstellbarkeit ganzer Zahlen durch Kaplanskys tern 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 and , Collection of articles dedicated to Derrick Henry Lehmer on the occasion of his seventieth birthday. Math. Comp. 29, 243269, 1975. MR 0457373 (56:15581a)
 10.
 G. Shimura, On modular forms of half integral weight. Ann. of Math. (2) 97, 440481, 1973. MR 0332663 (48:10989)
 11.
 K. Soundararajan, Moments of the Riemann zetafunction, Ann. of Math. (2) 170, 2009.
 12.
 H. M. Stark, The analytic theory of algebraic numbers. Bull. Amer. Math. Soc. 81, no. 6, 961972, 1975. MR 0444611 (56:2961)
 13.
 E. C. Titchmarsh, The Theory of the Riemann Zetafunction, 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
Email:
vchandee@math.stanford.edu
DOI:
http://dx.doi.org/10.1090/S0002993909100758
PII:
S 00029939(09)100758
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.
