ED Mathématiques et Informatique
Probabilistic distribution of prime numbers and ideas : Hooley's conjecture and Chebyshev's bias
by Mounir HAYANI (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 16h00 - Salle de conférence Institut de Mathématiques de Bordeaux, 351, cours de la Libération - bâtiment A 33 33 405 TALENCE
in front of the jury composed of
- Florent JOUVE - Professeur des universités - Institut de mathématiques de Bordeaux - Directeur de these
- Lucile DEVIN - Maîtresse de conférences - Université du Littoral Côte d'Opale - Examinateur
- Régis DE LA BRETèCHE - Professeur des universités - Université Paris cité - Rapporteur
- Nathan NG - Full professor - University of Lethbridge - Rapporteur
- Guillaume RICOTTA - Maître de conférences - Institut de mathématiques de Bordeaux - Examinateur
- Youness LAMZOURI - Professeur des universités - Université de Lorraine - Examinateur
Chebyshev's bias refers to the fact that primes tend to favor certain residue classes over others. This phenomenon was first observed by Chebyshev in the classical setting of prime numbers modulo $4$ and was later systematically studied by Rubinstein and Sarnak through the machinery of limiting distributions and zeros of Dirichlet $L$-functions. It was subsequently extended by Nathan Ng to the case of number fields and further investigated by several authors. In particular, Fiorilli and Jouve proved unconditionally that, contrary to the classical setting, extreme cases of Chebyshev's bias can occur in number fields. This result served as the starting point of this work. We first extend and generalize their unconditional results. Then, under some classical hypotheses, we establish a full characterization of these extreme cases of Chebyshev's bias. The second part of this thesis is devoted to the study of the variance of primes in arithmetic progressions. This measure of the fluctuations in the distribution of primes in arithmetic progressions has been studied by many authors. In the natural framework where one wants to evaluate the gaps between the prime counting function in an arithmetic progression and de la Vallée Poussin's asymptotics, Hooley conjectured an upper bound for the variance. Its range of validity has been subject to discussion. As a matter of fact, Fiorilli and Martin showed the conjecture fails if one does not restrict to a suitable range, demonstrating that the variance takes larger values than expected. We investigate whether a natural smoothing of the variance can restore the validity of Hooley's conjecture in a wider range. Our results show that even in the ``smooth setting'', this is not possible, reinforcing evidence for the presence of unexpected irregularities in the distribution of primes.
ED Sciences Chimiques
Development of new sulfate-containing positive electrode materials based on abundant elements for Na-ion batteries
by Anastasia GREBENSHCHIKOVA (ICMCB - Institut de Chimie de la Matière Condensée de Bordeaux)
The defense will take place at 14h00 - Amphithéâtre Institut de Chimie de la Matière Condensée de Bordeaux, ICMCB UMR 5026, F-33600 Pessac, France 87, Avenue du Docteur Schweitzer
in front of the jury composed of
- Laurence CROGUENNEC - Directrice de recherche - Université de Bordeaux - Directeur de these
- Prabeer BARPANDA - Associate Professor - Materials Research Centre, Indian Institute of Science, Bangalore, India - Rapporteur
- Marine REYNAUD - Directrice de recherche - CIC energiGUNE, Vitoria-Gasteiz, Spain - Rapporteur
- Christian MASQUELIER - Professeur des universités - Université de Picardie Jules Verne, France - CoDirecteur de these
- Cyril AYMONIER - Directeur de recherche - Université de Bordeaux - Examinateur
- Laure BERTRY - Ingénieure de recherche - Syensqo - Examinateur
- Olivier MENTRE - Directeur de recherche - Université de Lille - Examinateur
Scientific results of the PhD project, covered in this manuscript, include development of new sulfate and/or phosphate sodium iron positive electrode materials for Na-ion batteries. In particular, sodium iron sulfate Na2Fe3(SO4)4, a new compound, was obtained by ball milling synthesis method and preliminary electrochemical tests showed promising results. Additionally, operando X-ray diffraction and Mössbauer spectroscopy experiments, using synchrotron facilities, were performed in order to follow structural and redox mechanisms involved upon cycling. The next studied compounds belong to the family of mixed phosphate sulfate Na3-δFe2+β(PO4)y(SO4)3-y, showing alluaudite-type structure with different stoichiometries. They were obtained by solid-state method and for the first time, by ionothermal and mechanochemical syntheses. The solid-state synthesis, followed in situ by Synchrotron X-ray diffraction, showed formation of Na6Fe(SO4)4 and NaSICON-structured Na3-δFe2PO4(SO4)2 prior to stabilization of nearly pure alluaudite compound. This is the first evidence of any relationship between these structural types. Finally, in order to solve existing ambiguity of space group determination for the structural description of the NaSICON compound NaFe2PO4(SO4)2 it was synthesized by different synthetic routes and, for the pure sample, combination of electron, X-ray and neutron diffractions revealed the R-3 space group with a structural model characterized by disorder in one Na site.