Oscillatory integrals with uniformity in parameters
Eren Mehmet Kıral1; Ian Petrow2; Matthew P. Young1
1 Department of Mathematics Texas A&M University College Station TX 77843-3368, U.S.A.
2 ETH Zürich - Departement Mathematik HG G 66.4 Rämistrasse 101 8092 Zürich, Switzerland
Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 145-159.

We prove a sharp asymptotic formula for certain oscillatory integrals that may be approached using the stationary phase method. The estimates are uniform in terms of auxiliary parameters, which is crucial for application in analytic number theory.

Nous prouvons une formule asymptotique précise pour certains types d’intégrales oscillatoires que l’on peut traiter par la méthode de la phase stationnaire. Les estimations sont uniformes en termes de paramètres auxiliaires, ce qui est crucial pour les applications en théorie analytique des nombres.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1072
Classification: 41A60, 42A38
Keywords: Oscillatory integrals, Stationary phase

Eren Mehmet Kıral 1; Ian Petrow 2; Matthew P. Young 1

1 Department of Mathematics Texas A&M University College Station TX 77843-3368, U.S.A.
2 ETH Zürich - Departement Mathematik HG G 66.4 Rämistrasse 101 8092 Zürich, Switzerland
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{JTNB_2019__31_1_145_0,
     author = {Eren Mehmet K{\i}ral and Ian Petrow and Matthew P. Young},
     title = {Oscillatory integrals with uniformity in parameters},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {145--159},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     doi = {10.5802/jtnb.1072},
     mrnumber = {3994723},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1072/}
}
TY  - JOUR
AU  - Eren Mehmet Kıral
AU  - Ian Petrow
AU  - Matthew P. Young
TI  - Oscillatory integrals with uniformity in parameters
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2019
SP  - 145
EP  - 159
VL  - 31
IS  - 1
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1072/
DO  - 10.5802/jtnb.1072
LA  - en
ID  - JTNB_2019__31_1_145_0
ER  - 
%0 Journal Article
%A Eren Mehmet Kıral
%A Ian Petrow
%A Matthew P. Young
%T Oscillatory integrals with uniformity in parameters
%J Journal de théorie des nombres de Bordeaux
%D 2019
%P 145-159
%V 31
%N 1
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1072/
%R 10.5802/jtnb.1072
%G en
%F JTNB_2019__31_1_145_0
Eren Mehmet Kıral; Ian Petrow; Matthew P. Young. Oscillatory integrals with uniformity in parameters. Journal de théorie des nombres de Bordeaux, Volume 31 (2019) no. 1, pp. 145-159. doi : 10.5802/jtnb.1072. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1072/

[1] Valentin Blomer; Rizwanur Khan; Matthew P. Young Distribution of Maass of holomorphic cusp forms, Duke Math. J., Volume 162 (2013) no. 14, pp. 2609-2644 | DOI | Zbl

[2] John B. Conrey; Henryk Iwaniec The cubic moment of central values of automorphic L-functions, Ann. Math., Volume 151 (2000) no. 3, pp. 1175-1216 | DOI | MR | Zbl

[3] Sidney W. Graham; Grigori Kolesnik van der Corput’s method of exponential sums, London Mathematical Society Lecture Note Series, 126, Cambridge University Press, 1991 | MR | Zbl

[4] Lars Hörmander The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Classics in Mathematics, Springer, 2003 | Zbl

[5] Martin N. Huxley Area, lattice points, and exponential sums, London Mathematical Society Monographs. New Series, 13, Oxford University Press, 1996 | MR | Zbl

[6] Henryk Iwaniec; Emmanuel Kowalski Analytic number theory, Colloquium Publications, 53, American Mathematical Society, 2004 | MR | Zbl

[7] Eren Mehmet Kıral; Matthew P. Young The fifth moment of modular L-functions (2017) (https://arxiv.org/abs/1701.07507)

[8] Ian Petrow; Matthew P. Young A generalized cubic moment and the Petersson formula for newforms, Math. Ann., Volume 373 (2019) no. 1-2, pp. 287-353 | DOI | MR | Zbl

[9] Elias M. Stein Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, 1993 | MR | Zbl

[10] Maciej Zworski Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012 | MR | Zbl

Cited by Sources: