Pretentiousness in analytic number theory
Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 159-173.

Dans ce rapport, préparé spécialement pour les XXVième Journées Arithmétiques, nous décrivons, dans un travail commun avec K. Soundararajan et Antal Balog, comment nous avons développé la notion de “prétention” pour nous aider à mieux comprendre plusieurs questions au sein de la théorie analytique des nombres.

In this report, prepared specially for the program of the XXVième Journées Arithmétiques, we describe how, in joint work with K. Soundararajan and Antal Balog, we have developed the notion of “pretentiousness” to help us better understand several key questions in analytic number theory.

DOI : 10.5802/jtnb.664
Granville, Andrew 1

1 Départment de Mathématiques et Statistique Université de Montréal CP 6128 succ Centre-Ville Montréal, QC H3C 3J7, Canada
@article{JTNB_2009__21_1_159_0,
     author = {Granville, Andrew},
     title = {Pretentiousness in analytic number theory},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {159--173},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     doi = {10.5802/jtnb.664},
     mrnumber = {2537710},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.664/}
}
TY  - JOUR
AU  - Granville, Andrew
TI  - Pretentiousness in analytic number theory
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2009
SP  - 159
EP  - 173
VL  - 21
IS  - 1
PB  - Université Bordeaux 1
UR  - http://archive.numdam.org/articles/10.5802/jtnb.664/
DO  - 10.5802/jtnb.664
LA  - en
ID  - JTNB_2009__21_1_159_0
ER  - 
%0 Journal Article
%A Granville, Andrew
%T Pretentiousness in analytic number theory
%J Journal de théorie des nombres de Bordeaux
%D 2009
%P 159-173
%V 21
%N 1
%I Université Bordeaux 1
%U http://archive.numdam.org/articles/10.5802/jtnb.664/
%R 10.5802/jtnb.664
%G en
%F JTNB_2009__21_1_159_0
Granville, Andrew. Pretentiousness in analytic number theory. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 1, pp. 159-173. doi : 10.5802/jtnb.664. http://archive.numdam.org/articles/10.5802/jtnb.664/

[1] A. Balog, A. Granville, K. Soundararajan, Multiplicative Functions in Arithmetic Progressions (to appear). | MR

[2] E. Bombieri, Le grand crible dans la théorie analytique des nombres . Astérisque 18 (1987/1974). | Numdam | MR | Zbl

[3] D.A. Burgess, On character sums and L-series, I. Proc. London Math. Soc. 12 (1962), 193–206. On character sums and L-series, II . Proc. London Math. Soc. 13 (1963), 524–536. | MR | Zbl

[4] H. Davenport, Multiplicative number theory. Springer Verlag, New York, 1980. | MR | Zbl

[5] J.B. Friedlander, Selberg’s formula and Siegel’s zero. In Recent progress in analytic number theory, Vol. 1 (Durham, 1979), 15–23. Academic Press, London-New York, 1981. | MR | Zbl

[6] P.X. Gallagher, A large sieve density estimate near σ=1 . Invent. Math. 11 (1970), 329–339. | EuDML | MR | Zbl

[7] A. Granville, G. Martin, Prime Number Races. Amer. Math. Monthly 113 (2006), 1–33. | MR | Zbl

[8] A. Granville, K. Soundararajan, The Spectrum of Multiplicative Functions. Ann. of Math. 153 (2001), 407–470. | MR | Zbl

[9] A. Granville, K. Soundararajan, Large Character Sums. J. Amer. Math. Soc 14 (2001), 365–397. | MR | Zbl

[10] A. Granville, K. Soundararajan, Large Character sums: pretentious characters and the Pólya-Vinogradov theorem. J. Amer. Math. Soc. 20 (2007), 357–384. | MR

[11] A. Granville, K. Soundararajan, Large Character Sums: pretentious characters, Burgess’s theorem and the location of zeros (to appear). | MR

[12] G. Halász, On the distribution of additive and mean-values of multiplicative functions. Stud. Sci. Math. Hungar. 6 (1971), 211–233. | MR | Zbl

[13] G. Halász, On the distribution of additive arithmetic functions. Acta Arith. 27 (1975), 143–152. | MR | Zbl

[14] H. Iwaniec, E. Kowalski, Analytic number theory. Amer. Math. Soc., Providence, Rhode Island, 2004. | MR | Zbl

[15] H.L. Montgomery, R.C. Vaughan, Exponential sums with multiplicative coefficients. Invent. Math. 43 (1977), 69–82. | MR | Zbl

[16] R.E.A.C. Paley, A theorem on characters. J. London Math. Soc. 7 (1932), 28–32. | Zbl

[17] G. Pólya , Über die Verteilung der quadratischen Reste und Nichtreste. Göttingen Nachrichten (1918), 21–29.

[18] A. Selberg, An elementary proof of the prime number theorem for arithmetic progressions. Can. J. Math. 2 (1950), 66–78. | MR | Zbl

[19] I.M. Vinogradov, Über die Verteilung der quadratischen Reste und Nichtreste. J. Soc. Phys. Math. Univ. Permi 2 (1919), 1–14.

Cité par Sources :