# Forskjeller

Her vises forskjeller mellom den valgte versjonen og den nåværende versjonen av dokumentet.

Begge sider forrige revisjon Forrige revisjon Neste revisjon | Forrige revisjon | ||

ma3150:2019v:start [2019-03-15] seip [Oral presentations] |
ma3150:2019v:start [2019-04-12] (nåværende versjon) seip [Exam, dates and location] |
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The syllabus for the course is as defined by the lectures described in detail below. I expect you to be able to present the basic concepts and ideas discussed during the lectures. The exercises should be viewed as an integral part of the syllabus. | The syllabus for the course is as defined by the lectures described in detail below. I expect you to be able to present the basic concepts and ideas discussed during the lectures. The exercises should be viewed as an integral part of the syllabus. | ||

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+ | The final lecture took place on March 19. You are supposed to work on the topic for your oral presentation during the four remaining weeks of the semester. | ||

===== Reference Group ===== | ===== Reference Group ===== | ||

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* Lecture 19, March 12. We continue our discussion of primitive characters and Gauss sums, covering roughly 8.8 - 8.12 in Apostol. | * Lecture 19, March 12. We continue our discussion of primitive characters and Gauss sums, covering roughly 8.8 - 8.12 in Apostol. | ||

* Lecture 20, March 18. We establish the functional equation for \( L(s,\chi) \) when \( \chi \) is a primitive character, following Ch. 9 in Davenport. | * Lecture 20, March 18. We establish the functional equation for \( L(s,\chi) \) when \( \chi \) is a primitive character, following Ch. 9 in Davenport. | ||

- | * Lecture 21, March 19 (FINAL LECTURE). We recall our route to the prime number theorem and discuss how to proceed to obtain the corresponding result for arithmetic progressions. The lecture will include some words on zero-free regions (see Ch. 14 in Davenport). | + | * Lecture 21, March 19 (FINAL LECTURE). We recall our route to the prime number theorem and discuss how to proceed to obtain the corresponding result for arithmetic progressions. The lecture will include some words on zero-free regions and Siegel zeros (see Ch. 14 in Davenport). |

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As part of the oral exam, the students should give short presentations of topics assigned to them. These are topics that are not covered by the lectures. You may choose a topic yourself (to be approved by me) or choose one from the following list: | As part of the oral exam, the students should give short presentations of topics assigned to them. These are topics that are not covered by the lectures. You may choose a topic yourself (to be approved by me) or choose one from the following list: | ||

- | - Mertens's theorems and Mertens's constant | + | - Mertens's theorems and Mertens's constant (chosen by ''Tor Kringeland'') |

- The Bertrand--Chebyshev theorem, including Ramanujan and Erdős's work on it (chosen by ''Claudia Wohlgemuth'') | - The Bertrand--Chebyshev theorem, including Ramanujan and Erdős's work on it (chosen by ''Claudia Wohlgemuth'') | ||

- | - Ramanujan primes | + | - Ramanujan primes (chosen by ''Oskar Vikhamar-Sandberg'') |

- | - Skewes's number and sign changes in \( \pi(x)-\operatorname{li}(x) \) | + | - Skewes's number and sign changes in \( \pi(x)-\operatorname{li}(x) \) (chosen by ''Knut Bjarte Haus'') |

- General distribution of nontrivial zeros of \(\zeta(s)\) | - General distribution of nontrivial zeros of \(\zeta(s)\) | ||

- Zeros on the critical line, including density results (chosen by ''Terje Bull Karlsen'') | - Zeros on the critical line, including density results (chosen by ''Terje Bull Karlsen'') | ||

- The error term in the prime number theorem and zero-free regions | - The error term in the prime number theorem and zero-free regions | ||

- The Lindelöf hypothesis and the density hypothesis | - The Lindelöf hypothesis and the density hypothesis | ||

- | - Mean value theorems - results and conjectures | + | - Mean value theorems - results and conjectures (chosen by ''Fredrik Vaagen'') |

- Zeta functions for which RH fails | - Zeta functions for which RH fails | ||

- | - Dirichlet's divisor problem, including Voronoi's summation formula | + | - Dirichlet's divisor problem, including Voronoi's summation formula (chosen by ''Lars Magnus Øverlier'') |

- Elementary sieve methods and Brun's theorem on twin primes (chosen by ''Daniel Olaisen'') | - Elementary sieve methods and Brun's theorem on twin primes (chosen by ''Daniel Olaisen'') | ||

- Voronin's universality theorem and value distribution of the Riemann zeta function | - Voronin's universality theorem and value distribution of the Riemann zeta function | ||

- | - Lagarias's version of Guy Robin's criterion | + | - Lagarias's version of Guy Robin's criterion (chosen by Morten ''Ravnemyr'') |

- The Beurling--Nyman condition for RH | - The Beurling--Nyman condition for RH | ||

- Li's criterion for RH (chosen by ''William Tell'') | - Li's criterion for RH (chosen by ''William Tell'') | ||

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- Alternate proofs of the functional equation for \( \zeta(s)\) (Titchmarsh gives 7 proofs; take a look and make your own selection of some of them) | - Alternate proofs of the functional equation for \( \zeta(s)\) (Titchmarsh gives 7 proofs; take a look and make your own selection of some of them) | ||

- Approximations of \(\zeta(s)\), including the approximate functional equation | - Approximations of \(\zeta(s)\), including the approximate functional equation | ||

- | - The Riemann--Weil explicit formula. | + | - The Riemann--Weil explicit formula (chosen by ''Henrik Romnes'') |

+ | - Siegel zeros. | ||

The aim of the presentations is to convey to your peers what the topic is about and the most important and interesting problems and results associated with it. You are not expected to study proofs of deep theorems, but you should be able to say a little more than what we can find on the Wikipedia. You may choose to give a blackboard lecture or a Beamer/Powerpoint presentation. Each presentation should last for about 15--20 minutes. | The aim of the presentations is to convey to your peers what the topic is about and the most important and interesting problems and results associated with it. You are not expected to study proofs of deep theorems, but you should be able to say a little more than what we can find on the Wikipedia. You may choose to give a blackboard lecture or a Beamer/Powerpoint presentation. Each presentation should last for about 15--20 minutes. | ||

- | Please let me know your choice of topic before **March 29**. There should be only student per topic. Once a topic is chosen, I will make a note of it in the list above. Accordingly, we follow the principle of "first-come, first-served" when assigning topics. | + | Please let me know your choice of topic before **March 29**. There should be only one student per topic. Once a topic is chosen, I will make a note of it in the list above. Accordingly, we follow the principle of "first-come, first-served" when assigning topics. |

===== Exam, dates and location ===== | ===== Exam, dates and location ===== | ||

- | The oral presentations will be given on May 8. Oral examinations will take place on May 9. Both events will take place in Room 656 SB2. A detailed schedule will be announced in due course. | + | The oral presentations will be given on May 8. **You are strongly encouraged to be present at all the presentations!** Oral examinations will take place on May 9. Both events will take place in Room 656 SB2. A detailed schedule is as follows. |

- | ===== Guidance and consultation before the exam ===== | + | === Schedule May 8 === |

- | I will be available for consultation | + | * 08:30 - 08:55 ''Tor Kringeland'': Mertens's theorems and Mertens's constant |

+ | * 09:00 - 09:25 ''Claudia Wohlgemuth'': The Bertrand--Chebyshev theorem, including Ramanujan and Erdős's work on it | ||

+ | * 09:30 - 09:55 ''Oskar Vikhamar-Sandberg'': Ramanujan primes | ||

+ | * 10:00 - 10:25 ''Knut Bjarte Haus'': Skewes's number and sign changes in \( \pi(x)-\operatorname{li}(x) \) | ||

+ | * 10:30 - 10:55 ''Terje Bull Karlsen'': Zeros on the critical line, including density results | ||

+ | * 11:00 - 11:25 ''Lars Magnus Øverlier'': Dirichlet's divisor problem, including Voronoi's summation formula | ||

+ | * 11:30 - 11:55 ''Fredrik Vaagen'': Mean value theorems - results and conjectures | ||

+ | * 13:00 - 13:25 ''William Tell'': Li's criterion for RH | ||

+ | * 13:30 - 13:55 ''Morten Ravnemyr'': Lagarias's version of Guy Robin's criterion | ||

+ | * 14:00 - 14:25 ''Daniel Olaisen'': Elementary sieve methods and Brun's theorem on twin primes | ||

+ | * 14:30 - 14:55 ''Henrik Romnes'': The Riemann–Weil explicit formula. | ||

+ | | ||

+ | === Schedule May 9 === | ||

+ | | ||

+ | * 08:30 - 08:55 ''Tor Kringeland'' | ||

+ | * 09:00 - 09:25 ''Claudia Wohlgemuth'' | ||

+ | * 09:30 - 09:55 ''Oskar Vikhamar-Sandberg'' | ||

+ | * 10:00 - 10:25 ''Knut Bjarte Haus'' | ||

+ | * 10:30 - 10:55 ''Terje Bull Karlsen'' | ||

+ | * 11:00 - 11:25 ''Lars Magnus Øverlier'' | ||

+ | * 11:30 - 11:55 ''Fredrik Vaagen'' | ||

+ | * 13:00 - 13:25 ''William Tell'' | ||

+ | * 13:30 - 13:55 ''Morten Ravnemyr'' | ||

+ | * 14:00 - 14:25 ''Daniel Olaisen'' | ||

+ | * 14:30 - 14:55 ''Henrik Romnes'' | ||

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+ | ===== Guidance and consultation before the exam ===== | ||

- | * April 4 -- 6 | + | Before the Easter break, I will be available for consultation until April 3. I will be traveling April 4 -- 11, and will again be available on April 12. After the Easter break, I will be available only May 6 -- 7. |

- | * April 15 | + | |

- | * May 6 -- 7. | + | |

- | You may in principle come at "any" time during these days, but I would recommend that you contact me in advance to make an appointment. | + | You may in principle come at "any" time during the days I am in my office, but I would recommend that you contact me in advance to make an appointment. |