Chapter 1 Primes Represented by Positive

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Chapter 1 Primes Represented by Positive Research Collection Doctoral Thesis Primes of the shape x² + ny² The distribution on average and prime number races Author(s): Ditchen, Jakob J. Publication Date: 2013 Permanent Link: https://doi.org/10.3929/ethz-a-010138958 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library Diss. ETH No. 21502 Primes of the shape x2 + ny2 The Distribution on Average and Prime Number Races A dissertation submitted to ETH Zürich for the degree of Doctor of Sciences Presented by Jakob Johann Ditchen Dipl.-Math. techn. Universität Karlsruhe (TH) Certificate of Advanced Study in Mathematics University of Cambridge Born on 10 November 1982 Citizen of Germany Accepted on the recommendation of Prof. Dr. Emmanuel Kowalski Examiner Prof. Dr. Özlem Imamoglu Co-examiner Prof. Dr. Philippe Michel Co-examiner 2013 Abstract This thesis focuses on uniformities and discrepancies in the distribution of prime numbers represented by positive definite integral binary quadratic forms of various discriminants. We prove results of Bombieri–Vinogradov and Barban–Davenport–Halberstam type on the average distribution of the primes with respect to their representability by these forms. Our results imply that the corresponding prime number theorem holds uniformly and with a non-trivial error term for almost all negative fundamental discriminants in long ranges. Moreover, we investigate a variant of “Chebyshev’s bias” between primes of the shapes x2 + ny2 and x2 + my2 for certain distinct positive integers n and m. Deutsche Zusammenfassung Die vorliegende Dissertation befasst sich mit Gleichmäßigkeiten und Diskrepanzen in der Verteilung von Primzahlen, die durch positiv definite, ganzzahlige binäre quadratische Formen mit unterschiedlichen Diskriminanten darstellbar sind. Wir beweisen Varianten des Satzes von Bombieri-Vinogradov sowie des Satzes von Barban-Davenport-Halberstam und zeigen so, dass der betreffende Primzahlsatz, von höchstens „wenigen“ Ausnahmen abgesehen, für negative Fundamentaldiskrimi- nanten in langen Intervallen ein gleichmäßiges und nicht-triviales Restglied aufweist. Des Weiteren untersuchen wir für gewisse Paare (n, m) positiver ganzer Zahlen eine Diskrepanz zwischen den Verteilungen von Primzahlen der Form x2 + ny2 und solchen der Form x2 + my2; dies stellt ein Gegenstück zu einer klassischen Beobach- tung von Tschebyscheff bezüglich Primzahlen der Formen 4k + 1 und 4k + 3 dar, die in den letzten Jahren intensiv untersucht wurde. Binäre quadratische Formen, das heißt homogene Polynome der Gestalt 2 2 f(x, y) = ax + bxy + cy (x, y ∈ Z) mit ganzzahligen Koeffizienten a, b und c, sind neben arithmetischen Folgen die einfachsten Polynome von denen bekannt ist, dass jedes von ihnen unendlich viele Primzahlen darzustellen vermag, sofern dem nicht Kongruenzbeziehungen der Koeffizienten offensichtlich entgegenstehen. Die analytische Theorie der Darstellung von Primzahlen durch fest gewählte binäre quadratische Formen ist ähnlich gut erforscht wie jene der Primzahlen in fest gewählten arithmetischen Folgen. Es ist hingegen nur wenig darüber bekannt, wie sich diese Eigenschaften im Durchschnitt über mehrere binäre quadratische Formen unterschiedlicher Diskriminante verhalten oder wie sie im Vergleich zweier verschiedener Formen von einander abweichen – andererseits existieren zahlreiche solcher Resultate für Primzahlen in arithmetischen Folgen. Für Primzahlen in arithmetischen Folgen wurden für die erstgenannte Art von Problemen mittels des Großen Siebs ab den 1960er Jahren Resultate erzielt, die es in Anwendungen häufig ermöglichen, auf den Gebrauch der verallgemeinerten Riemannschen Vermutung zu verzichten. Die bekanntesten dieser Ergebnisse sind der Satz von Bombieri-Vinogradov sowie der Satz von Barban-Davenport-Halberstam. Diese zeigen zum einen, dass das Restglied im Primzahlsatz für arithmetische Folgen im Durchschnitt dem durch die Riemannsche Vermutung vorhergesagten entspricht; dabei wird der Durchschnitt über die Moduln der arithmetischen Folgen im selben Bereich betrachtet, in welchem die Riemannsche Vermutung nicht-triviale Ergebnisse liefert. Zum anderen konnte gezeigt werden, dass der mittlere quadratische Fehler im Primzahlsatz sehr klein ist, wenn sowohl über die Moduln als auch über deren Restklassen gemittelt wird; der hier zulässige Bereich für die Moduln übersteigt dabei sogar den durch die Riemannsche Vermutung kontrollierten Bereich. In der vorliegenden Arbeit werden analoge Resultate für positiv definite, ganzzahlige binäre quadratische Formen gefunden: Sei X eine große, positive Zahl. Für die Anzahl der Primzahlen p 6 X, für welche – bei gegebener ganzer Zahl n – ganze Zahlen x und y existieren, so dass p sich in der Form p = x2 + ny2 schreiben lässt, zeigen wir insbesondere, dass der zugehörige Primzahlsatz für die quadratfreien, positiven ganzen Zahlen n ≡ 1 (mod 4) unterhalb von etwa X1/10 gleichmäßig in n gilt – abgesehen von höchstens „wenigen“ Ausnahmen. Allgemeiner beweisen wir konkret, dass für alle A > 0 eine Konstante B = B(A) existiert, so dass für alle ε > 0 die Beziehung X0 li(X) 1/2 −A max π(X; q, C) − ε,A Q X(log X) C∈K(q) e(C)h(q) q>−Q 10+ε −B gilt, falls Q 6 X(log X) ist. Hierbei bezeichnet π(X; q, C) die Anzahl der Primzahlen p 6 X, welche durch die quadratischen Formen der Formenklasse C aus der Formenklassen- gruppe K(q) zur Diskriminante q darstellbar sind, h(q) bezeichnet die Klassenzahl zu dieser Diskriminante, li steht für das logarithmische Integral und e(C) ist eine von der Klasse abhän- gige Konstante; die Summe auf der linken Seite läuft über negative Fundamentaldiskriminanten q > −Q mit q 6≡ 0 (mod 8). Ferner zeigen wir, dass das Restglied im Primzahlsatz für positiv definite binäre quadra- tische Formen im quadratischen Mittel über sowohl Fundamentaldiskriminanten als auch die zugehörigen Formenklassen in einem größeren Bereich klein ist: Für alle A > 0 existiert eine Konstante B = B(A), so dass für alle ε > 0 die Beziehung 2 X0 X li(X) 1/2 2 −A π(X; q, C) − ε,A Q X (log X) e(C)h(q) q>−Q C∈K(q) 3+ε −B gilt, falls Q 6 X(log X) ist. Beide Ergebnisse erreichen nicht die Stärke der oben genannten Resultate für arithmetische Folgen. Dies ist unter anderem auf den Umstand zurückzuführen, dass es uns lediglich gelingt eine schwächere Version einer Ungleichung zum Großen Sieb für komplexe Klassengruppen- charaktere zu finden, welche für Ergebnisse dieses Typs unentbehrlich scheint. Während die bisher genannten Ergebnisse sich mit der Untersuchung von Uniformität in der Verteilung von durch arithmetische Folgen respektive binäre quadratische Formen darstellba- ren Primzahlen befassen, ist die Frage nach Diskrepanzen in diesen Verteilungen nicht minder interessant. Tschebyscheff bemerkte bereits, dass die Anzahl der Primzahlen in der Folge 4k + 1 unterhalb einer gegebenen Zahl meist kleiner ist als diejenige in der Folge 4k + 3. Dem Prim- zahlsatz zufolge sind beide Anzahlen asymptotisch gleich, so dass die Ursachen dieser „Vorliebe“ der Primzahlen für die zweite Folge nicht offensichtlich sind. Erst in den letzten Jahren wurde diese Diskrepanz in allgemeiner Form für arithmetische Folgen eingehend untersucht. Wir untersuchen in dieser Arbeit einen ähnlichen Effekt, der sich für die Anzahl der Prim- zahlen der Form x2 + ny2 und solche der Form x2 + my2 unterhalb einer gegebenen Zahl offenbart, wenn sich die zugehörigen Diskriminanten der beiden Formen in der Klassenzahl zwar gleichen – und somit, nach dem Primzahlsatz, auch das asymptotische Verhalten der Verteilun- gen übereinstimmt –, sie sich in der Anzahl ihrer ungeraden Primfaktoren aber unterscheiden. Von den großen Meistern wie Fermat, Euler, Gauß und Dirichlet wurde Primzahlen der Form x2 + ny2 mindestens ebenso viel Beachtung geschenkt wie Primzahlen der Form a + nk. Und noch de la Vallée Poussin bewies in seiner Arbeit zum Primzahlsatz diesen nicht nur in der gewöhnlichen Form und in der Form für arithmetische Folgen, sondern ebenfalls gleich in der Form für positiv definite binäre quadratische Formen. Wiewohl Primzahlen, die durch binäre quadratische Formen darstellbar sind, seither immer wieder prominent in Erscheinung getreten sind – beispielsweise als wichtiger Bestandteil bestimmter Faktorisierungsalgorithmen –, haben die Untersuchungen bezüglich Gleichmäßigkeiten und Diskrepanzen in ihrer Verteilung jedoch bei weitem nicht mehr dieselbe Aufmerksamkeit erhalten wie die entsprechende Forschung zu Primzahlen in arithmetischen Folgen, die sich häufig einfacher gestaltet. Die vorliegende Arbeit möchte einen Beitrag dazu leisten, diese Lücke dereinst zu schließen. Meinen Eltern gewidmet Contents Primes of the shape x2 + ny2: The Distribution on Average and Prime Number Races Preface 1 Notation 5 1 Primes represented by positive definite binary quadratic forms 7 1.1 The composition of binary quadratic forms and form classes . 9 1.2 Algebraic methods for arithmetic objects . 11 1.3 The Chebotarev density theorem and conditional results . 17 2 The average distribution of primes represented by positive definite binary quadratic forms with varying discriminant 21 2.1 Mean-value results for primes in arithmetic progressions . 22 2.2 A large sieve inequality for complex ideal class group characters . 25 2.3 Results of Bombieri–Vinogradov type . 34 2.4 The mean square distribution . 57 2.5 Applications and open questions . 63 3 Chebyshev’s bias and prime number races for binary quadratic forms 71 3.1 Bias in
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