240075 Analytische Zahlentheorie (V) (SoSe 2024)
Contents, comment
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Ein Höhepunkt der Veranstaltung ist der Beweis des Primzahlsatzes, der besagt, dass der Anteil der Primzahlen unter den ersten n natürlichen Zahlen etwa gleich dem Kehrwert des natürlichen Logarithmus von n ist, wobei der prozentuale Fehler mit wachsendem n gegen Null geht.
Dieser Satz wird mit Hilfe der Riemannschen Zetafunktion
ζ(s) = 1-s + 2-s + 3-s + . . .
bewiesen, wobei zunehmend verfeinerte Methoden aus der Funktionentheorie jeweils bessere Fehlerschranken ermöglichen. Die bestmöglichen Fehlerschranken gelten, falls die Riemannsche Vermutung richtig ist.
Weitere Themen sind die Verteilung von Primzahlen in arithmetischen Folgen, die Werte der Riemannschen Zetafunktion für ganzzahlige s<0 und die Dedekindsche Zetafunktion quadratischer Zahlkörper.
Requirements for participation, required level
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Funktionentheorie
Bibliography
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- D. J. Newman, Analytic number theory. Springer-Verlag 1998
- J. Brüdern, Einführung in die analytische Zahlentheorie. Springer-Verlag 1995
- T. M. Apostol, Introduction to analytic number theory. Springer-Verlag 1976
- S. J. Patterson, An introduction to the theory of the Riemann zeta function. Cambridge University Press 1988
- A.E. Ingham, The distribution of prime numbers. Cambridge Univ. Press 2000
Teaching staff
Dates ( Calendar view )
| Frequency | Weekday | Time | Format / Place | Period | |
|---|---|---|---|---|---|
| weekly | Mo | 8-10 | X-E0-226 | 08.04.-19.07.2024
not on: 5/20/24 |
|
| weekly | Do | 8-10 | V4-112 | 08.04.-19.07.2024
not on: 5/9/24 / 5/30/24 |
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