Pretentiousness in analytic number theory

Andrew Granville[1]

  • [1] Départment de Mathématiques et Statistique Université de Montréal CP 6128 succ Centre-Ville Montréal, QC H3C 3J7, Canada

Journal de Théorie des Nombres de Bordeaux (2009)

  • Volume: 21, Issue: 1, page 159-173
  • ISSN: 1246-7405

Abstract

top
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.

How to cite

top

Granville, Andrew. "Pretentiousness in analytic number theory." Journal de Théorie des Nombres de Bordeaux 21.1 (2009): 159-173. <http://eudml.org/doc/10868>.

@article{Granville2009,
abstract = {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.},
affiliation = {Départment de Mathématiques et Statistique Université de Montréal CP 6128 succ Centre-Ville Montréal, QC H3C 3J7, Canada},
author = {Granville, Andrew},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {pretentiousness; prime number theorem; character sums; mean values of totally multiplicative functions; multiplicative functions in arithmetic progressions},
language = {eng},
number = {1},
pages = {159-173},
publisher = {Université Bordeaux 1},
title = {Pretentiousness in analytic number theory},
url = {http://eudml.org/doc/10868},
volume = {21},
year = {2009},
}

TY - JOUR
AU - Granville, Andrew
TI - Pretentiousness in analytic number theory
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2009
PB - Université Bordeaux 1
VL - 21
IS - 1
SP - 159
EP - 173
AB - 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.
LA - eng
KW - pretentiousness; prime number theorem; character sums; mean values of totally multiplicative functions; multiplicative functions in arithmetic progressions
UR - http://eudml.org/doc/10868
ER -

References

top
  1. A. Balog, A. Granville, K. Soundararajan, Multiplicative Functions in Arithmetic Progressions (to appear). Zbl1306.11078MR1691308
  2. E. Bombieri, Le grand crible dans la théorie analytique des nombres . Astérisque 18 (1987/1974). Zbl0292.10035MR891718
  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. Zbl0106.04004MR132733
  4. H. Davenport, Multiplicative number theory. Springer Verlag, New York, 1980. Zbl0453.10002MR606931
  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. Zbl0459.10029MR637340
  6. P.X. Gallagher, A large sieve density estimate near σ = 1 . Invent. Math. 11 (1970), 329–339. Zbl0219.10048MR279049
  7. A. Granville, G. Martin, Prime Number Races. Amer. Math. Monthly 113 (2006), 1–33. Zbl1139.11037MR2202918
  8. A. Granville, K. Soundararajan, The Spectrum of Multiplicative Functions. Ann. of Math. 153 (2001), 407–470. Zbl1036.11042MR1829755
  9. A. Granville, K. Soundararajan, Large Character Sums. J. Amer. Math. Soc 14 (2001), 365–397. Zbl0983.11053MR1815216
  10. A. Granville, K. Soundararajan, Large Character sums: pretentious characters and the Pólya-Vinogradov theorem. J. Amer. Math. Soc. 20 (2007), 357–384. Zbl1210.11090MR2276774
  11. A. Granville, K. Soundararajan, Large Character Sums: pretentious characters, Burgess’s theorem and the location of zeros (to appear). Zbl1210.11090MR1815216
  12. G. Halász, On the distribution of additive and mean-values of multiplicative functions. Stud. Sci. Math. Hungar. 6 (1971), 211–233. Zbl0226.10046MR319930
  13. G. Halász, On the distribution of additive arithmetic functions. Acta Arith. 27 (1975), 143–152. Zbl0256.10028MR369292
  14. H. Iwaniec, E. Kowalski, Analytic number theory. Amer. Math. Soc., Providence, Rhode Island, 2004. Zbl1059.11001MR2061214
  15. H.L. Montgomery, R.C. Vaughan, Exponential sums with multiplicative coefficients. Invent. Math. 43 (1977), 69–82. Zbl0362.10036MR457371
  16. R.E.A.C. Paley, A theorem on characters. J. London Math. Soc. 7 (1932), 28–32. Zbl0003.34101
  17. G. Pólya , Über die Verteilung der quadratischen Reste und Nichtreste. Göttingen Nachrichten (1918), 21–29. Zbl46.0265.02
  18. A. Selberg, An elementary proof of the prime number theorem for arithmetic progressions. Can. J. Math. 2 (1950), 66–78. Zbl0036.30605MR33306
  19. I.M. Vinogradov, Über die Verteilung der quadratischen Reste und Nichtreste. J. Soc. Phys. Math. Univ. Permi 2 (1919), 1–14. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.