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The Laurent-Stieltjes Constants for the Zeta Function

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The Laurent-Stieltjes Constants for the Zeta Function Thèse de Sumaia Saad Eddin, Lille 1, 2013 No d’ordre : 41113 THESIS In order to become DOCTOR FROM UNIVERSITÉ LILLE 1 DOCTORAL SCHOOL:SCIENCES POUR L’ INGÉNIEUR NO 72 LABORATOIRE PAUL PAINLEVÉ MATHEMATICS by Sumaia SAAD EDDIN ON TWO PROBLEMS CONCERNING THE LAURENT-STIELTJES COEFFICIENTS OF DIRICHLET L-SERIES Supervised by Olivier RAMARÉ Defended on Monday 10th June, 2013 Dissertation committee: Recorder Andrew BOOKER Professor, University of Bristol, Bristol, UK Examiner Driss ESSOUABRI Professor, Université Jean Monnet, S. Étienne, France Examiner Bruno MARTIN Maître de Conférence, Université de Littoral, France Examiner Kohji MATSUMOTO Professor, Nagoya University , Japan Recorder Francesco PAPPALARDI Professor, Università Roma Tre, Roma, Italy Recorder Y. François PÉTERMANN Professor, University de Genève, Suisse Examiner Hervé QUEFFÉLEC Professor, Université Lille 1, Lille, France Supervisor Olivier RAMARÉ Chargé de recherches CNRS, Université Lille 1, France © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 Abstract In this thesis, we give explicit upper bound for the Laurent-Stieltjes constants in the following two cases: 1. The character χ is fixed and the order n goes to infinity. 2. The order n is 0 and the conductor q goes to infinity. The Laurent-Stieltjes constants γn(χ) are the coefficients of the expansion in the Lau- rent series of the Dirichlet L-series. This thesis is divided to three parts: In first part of this thesis, we study the asymptotic behavior in n of these con- stants. In 1985, Matsuoka gave an excellent asymptotic expansion for these constants in the case of the zeta function, We extend the formula of Matsuoka to the Dirichlet + γ χ ≤ < π e(n 1)/2 L- functions. We give an upper bound for n( ) when 1 q 2 n+1 . In the case when χ = χ0 and q = 1, this leads to a sizeable improvement of Matsuoka’s bound and on previous results. By using this result, we also deduce an approximation of the Dirichlet L-functions in the neighborhood of z = 1 by a short Taylor polynomial. We provide a direct proof of Matsuoka’s formula that avoids complex analysis and, more importantly, the functional equation of ζ(z)andL(z,χ). The second part of this thesis deals more specifically with the first Laurent-Stieltjes coefficient, namely L(1,χ). In 2001, Ramaré gave an approximate formula for L(1,χ) depending on Fourier transforms. Thanks to this formula, he obtained the best ex- plicit upper bound for L(1,χ) to date. Here, we study the most difficult case when χ = (2) 1, and get an improvement. As application, we deduce an explicit upper bound for the class number of any real quadratic field Q q , improving on a result by Le. In the last part, we follow the method of Ramaré for giving an upper bound of the first Laurent-Stieltjes coefficient but this time when χ(3) = 0. This result is an improvement on a similar one due to Louboutin. © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 Résumé (French version) Dans cette thèse, nous donnons des majorations explicites pour les constantes de Laurent-Stieltjes dans les deux cas suivants : 1. Le caractère χ est fixé et n tend vers l’infini. 2. L’ordre n est égal à 0 et le conducteur q tend vers l’infini. Les constantes de Laurent-Stieltjes γn(χ) sont les coefficients qui interviennent dans le développement en série de Laurent des séries L de Dirichlet. Cette thèse est com- posée de trois parties : Dans la première partie de cette thèse, nous étudions le comportement asympto- tique en n de ces constantes. En 1985, Matsuoka a donné un développement asymp- totique des constantes de Laurent-Stieltjes pour la fonction zêta de Riemann. Ici, nous prolongeons la formule de Matsuoka aux fonctions L de Dirichlet. Nous don- + γ χ ≤ < π e(n 1)/2 χ = χ = nons une majoration de n( ) pour 1 q 2 n+1 . Dans le cas où 0 et q 1, notre résultat donne une réelle amélioration de la majoration de Matsuoka et des ré- sultats précédents. En utilisant cette majoration, nous déduisons aussi une approxi- mation des fonctions L de Dirichlet au voisinage de z = 1 par un polynôme de Taylor relativement court. En outre, nous donnons une preuve directe de la formule de Mat- suoka en évitant l’analyse complexe et de façon notoire l’équation fonctionnelle de la fonction zêta de Riemann et des fonctions L de Dirichlet. Dans la deuxième partie de cette thèse, nous nous intéressons au premier coef- ficient de Laurent-Stieltjes L(1,χ). En 2001, Ramaré a donné une formule approchée pour L(1,χ) en utilisant des transformées de Fourier. Grâce à cette formule, Ramaré a obtenu la meilleure majoration explicite pour L(1,χ) à ce jour. Nous étudions le cas le plus difficile lorsque χ(2) = 1 et obtenons une amélioration. Nous en déduisons aussi une majoration explicite pour le nombre de classes pour les corps quadratiques réels Q q . Cette majoration est une amélioration d’un résultat de Le. Dans la dernière partie, nous suivons la méthode de Ramaré pour donner une majoration explicite de L(1,χ) dans le cas où χ(3) = 0, améliorons un résultat de Louboutin. © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 Keywords: Dirichlet characters, Functional equation, Gauss sums, Gamma function, Matsuoka’s formula, Riemann zeta function, Laurent-Stieltjes coefficients, Dirichlet L-functions, Euler’s constant, class number, Euler-Maclaurin formula. Mots clés : Caractère de Dirichlet, équation fonctionnelle, somme de Gauss, fonction Gamma, formule de Matsuoka, fonction zêta de Riemann, coefficients de Laurent-Stieltjes, fonction L de Dirichlet, constante d’Euler, nombre de classes, formule d’Euler-Maclaurin. © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 This thesis has been prepared at Laboratoire Paul Painlevé Université Lille1-Sciences et Technologies Cité scientifique, BAT: M2 59655 Villeneuve D’ascq cédex +33 (0)3 20 43 48 50 +33 (0)3 20 43 43 02 [email protected] Site http://math.univ-lille1.fr © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 I dedicate this thesis to my father Mohammed, my mother Aida, my sisters: Nada, Hoda, Nibal and Rola, my brother Jamal and my love Waseem, for their constant support and unconditional love. I love you all. © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 Science sans conscience n’est que ruine de l’âme François RABELAIS © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 Acknowledgment It would not have been able to write this doctoral thesis without the help and support of the kind people around me, to only some of whom it is possible to give particular mention here. Above all, I would like to thank my supervisor, OLIVIER RAMARÉ, for his support on this thesis. Olivier helped me every step of the way; suggesting problems and av- enues of attack, steering my ideas into fruitful avenues, teaching me techniques and helping me improve my mathematical writing. I would not have been able to do this without him. Also, I consider myself very fortunate to be given an opportunity to work with such a talented mathematician. I would also like to thank my examiners, Professor ANDREW BOOKER, Professor FRANCESCO PAPPALARDI and Professor Y. FRANÇOIS PÉTERMANN, who provided en- couraging and constructive feedback. It is not an easy task, reviewing a thesis, and I am grateful for their thoughtful and detailed comments. Thank you for helping to shape and guide the direction of the work with your careful and instructive com- ments. I would like to thank my committee members, Professor DRISS ESSOUABRI,Pro- fessor BRUNO MARTIN, Professor HERVÉ QUEFFÉLEC for accepting to be in my jury. Special thanks are owed to Professor YASUSHI MATSUOKA for accepting to read my thesis whose first part depends on his previous work and to KOHJI MATSUMOTO for his support, it has been a pleasure discussing mathematics with such a respectful mathematician. I would like to thank DAVID PLATT from Bristol University for his contribution in the last part in this thesis, although I had never met him in person. © 2013 Tous droits réservés. http://doc.univ-lille1.fr Thèse de Sumaia Saad Eddin, Lille 1, 2013 xvi I would like to acknowledge the financial, academic and technical support to the University of Damascus, Université Lille 1 and its staff, that provided the necessary financial support for this research. I would like to thank my future husband Waseem for his personal support and great patience at all times. My parents, brother and sisters have given me their end- less support throughout, as always, for which my mere expression of thanks likewise does not suffice.
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