Oscillatory integrals with uniformity in parameters
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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 : 2018-03-03
Accepted : 2018-09-25
Published online : 2019-07-29
DOI : https://doi.org/10.5802/jtnb.1072
Classification:  41A60,  42A38
Keywords: Oscillatory integrals, Stationary phase
@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},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {31},
     number = {1},
     year = {2019},
     pages = {145-159},
     doi = {10.5802/jtnb.1072},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2019__31_1_145_0/}
}
Kıral, Eren Mehmet; Petrow, Ian; Young, Matthew P. 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/item/JTNB_2019__31_1_145_0/

[1] Valentin Blomer; Rizwanur Khan; Matthew P. Young Distribution of Maass of holomorphic cusp forms, Duke Math. J., Tome 162 (2013) no. 14, pp. 2609-2644 | Zbl 1312.11028

[2] John B. Conrey; Henryk Iwaniec The cubic moment of central values of automorphic L-functions, Ann. Math., Tome 151 (2000) no. 3, pp. 1175-1216 | Zbl 0973.11056

[3] Sidney W. Graham; Grigori Kolesnik van der Corput’s method of exponential sums, London Mathematical Society Lecture Note Series, Tome 126, Cambridge University Press, 1991 | Zbl 0713.11001

[4] Lars Hörmander The analysis of linear partial differential operators. I. Distribution theory and Fourier analysis, Classics in Mathematics, Springer, 2003 | Zbl 1028.35001

[5] Martin N. Huxley Area, lattice points, and exponential sums, London Mathematical Society Monographs. New Series, Tome 13, Oxford University Press, 1996 | Zbl 0861.11002

[6] Henryk Iwaniec; Emmanuel Kowalski Analytic number theory, Colloquium Publications, Tome 53, American Mathematical Society, 2004 | Zbl 1059.11001

[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., Tome 373 (2019) no. 1-2, pp. 287-353 | Zbl 07051745

[9] Elias M. Stein Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, Tome 43, Princeton University Press, 1993 | Zbl 0821.42001

[10] Maciej Zworski Semiclassical analysis, Graduate Studies in Mathematics, Tome 138, American Mathematical Society, 2012 | Zbl 1252.58001