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H 1 Large character sums

Auteurs : Lamzouri, Youness (Auteur de la Conférence)
CIRM (Editeur )

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    Résumé : For a non-principal Dirichlet character $\chi$ modulo $q$, the classical Pólya-Vinogradov inequality asserts that
    $M (\chi) := \underset{x}{max}$$| \sum_{n \leq x}$$\chi(n)| = O (\sqrt{q} log$ $q)$.
    This was improved to $\sqrt{q} log$ $log$ $q$ by Montgomery and Vaughan, assuming the Generalized Riemann hypothesis GRH. For quadratic characters, this is known to be optimal, owing to an unconditional omega result due to Paley. In this talk, we shall present recent results on higher order character sums. In the first part, we discuss even order characters, in which case we obtain optimal omega results for $M(\chi)$, extending and refining Paley's construction. The second part, joint with Alexander Mangerel, will be devoted to the more interesting case of odd order characters, where we build on previous works of Granville and Soundararajan and of Goldmakher to provide further improvements of the Pólya-Vinogradov and Montgomery-Vaughan bounds in this case. In particular, assuming GRH, we are able to determine the order of magnitude of the maximum of $M(\chi)$, when $\chi$ has odd order $g \geq 3$ and conductor $q$, up to a power of $log_4 q$ (where $log_4$ is the fourth iterated logarithm).

    Codes MSC :
    11L40 - Estimates on character sums
    11M06 - $ \zeta (s)$ and $L(s, \chi)$
    11N13 - Primes in progressions
    11N37 - Asymptotic results on arithmetic functions

      Informations sur la Vidéo

      Réalisateur : Hennenfent, Guillaume
      Langue : Français
      Date de publication : 01/06/17
      Date de captation : 25/05/17
      Collection : Research talks ; Number Theory
      Format : MP4
      Durée : 00:44:58
      Domaine : Number Theory
      Audience : Chercheurs ; Doctorants , Post - Doctorants
      Download : https://videos.cirm-math.fr/2017-05-25_Lamzouri.mp4

    Informations sur la rencontre

    Nom de la rencontre : Prime numbers and automatic sequences: determinism and randomness / Nombres premiers et suites automatiques : aléa et déterminisme
    Organisateurs de la rencontre : Dartyge, Cécile ; Drmota, Michael ; Martin, Bruno ; Mauduit, Christian ; Rivat, Joël ; Stoll, Thomas
    Dates : 22/05/17 - 26/05/17
    Année de la rencontre : 2017
    URL Congrès : http://conferences.cirm-math.fr/1595.html

    Citation Data

    DOI : 10.24350/CIRM.V.19171903
    Cite this video as: Lamzouri, Youness (2017). Large character sums. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19171903
    URI : http://dx.doi.org/10.24350/CIRM.V.19171903


    Voir aussi

    Bibliographie

    1. Lamzouri, Y., & Mangerel, A.P. (2017). Large odd order character sums and improvements of the Pólya-Vinogradov inequality. - https://arxiv.org/abs/1701.01042

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