Forskjeller
Her vises forskjeller mellom den valgte versjonen og den nåværende versjonen av dokumentet.
Begge sider forrige revisjon Forrige revisjon Neste revisjon | Forrige revisjon Neste revisjon Begge sider neste revisjon | ||
ma3150:2019v:start [2019-03-13] seip [Exercises] |
ma3150:2019v:start [2019-04-12] 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 |
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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' | + | - Mertens' |
- The Bertrand--Chebyshev theorem, including Ramanujan and Erdős' | - The Bertrand--Chebyshev theorem, including Ramanujan and Erdős' | ||
- | - Ramanujan primes | + | - Ramanujan primes |
- | - Skewes' | + | - Skewes' |
- 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 '' | - Zeros on the critical line, including density results (chosen by '' | ||
- 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 |
- Zeta functions for which RH fails | - Zeta functions for which RH fails | ||
- | - Dirichlet' | + | - Dirichlet' |
- Elementary sieve methods and Brun's theorem on twin primes (chosen by '' | - Elementary sieve methods and Brun's theorem on twin primes (chosen by '' | ||
- Voronin' | - Voronin' | ||
- | - Lagarias' | + | - Lagarias' |
- The Beurling--Nyman condition for RH | - The Beurling--Nyman condition for RH | ||
- | - Li's criterion for RH | + | - Li's criterion for RH (chosen by '' |
- The Bohr--Cahen formulas for abscissas of convergence and the growth of \(\sum_{n\le x} \mu(n)\). | - The Bohr--Cahen formulas for abscissas of convergence and the growth of \(\sum_{n\le x} \mu(n)\). | ||
- 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 \(\zeta(s)\), | - Approximations of \(\zeta(s)\), | ||
- | - The Riemann--Weil explicit formula. | + | - The Riemann--Weil explicit formula |
+ | - 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/ | 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/ | ||
- | 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, | + | 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, |
===== 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 | + | The oral presentations will be given on May 8. **You are strongly encouraged to be present at all the presentations!** |
- | ===== Guidance and consultation before the exam ===== | + | === Schedule May 8 === |
- | I will be available | + | * 08:30 - 08:55 '' |
+ | * 09:00 - 09:25 '' | ||
+ | * 09:30 - 09:55 '' | ||
+ | * 10:00 - 10:25 '' | ||
+ | * 10:30 - 10:55 '' | ||
+ | * 11:00 - 11:25 '' | ||
+ | * 11:30 - 11:55 '' | ||
+ | * 13:00 - 13:25 '' | ||
+ | * 13:30 - 13:55 '' | ||
+ | * 14:00 - 14:25 '' | ||
+ | * 14:30 - 14:55 '' | ||
+ | |||
+ | === Schedule May 9 === | ||
+ | |||
+ | * 08:30 - 08:55 '' | ||
+ | * 09:00 - 09:25 '' | ||
+ | * 09:30 - 09:55 '' | ||
+ | * 10:00 - 10:25 '' | ||
+ | * 10:30 - 10:55 '' | ||
+ | * 11:00 - 11:25 '' | ||
+ | * 11:30 - 11:55 '' | ||
+ | * 13:00 - 13:25 '' | ||
+ | * 13:30 - 13:55 '' | ||
+ | * 14:00 - 14:25 '' | ||
+ | * 14:30 - 14:55 '' | ||
+ | |||
+ | |||
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+ | ===== Guidance and consultation | ||
- | * April 4 -- 6 | + | Before the Easter break, I will be available for consultation until April 3. I will be traveling |
- | * April 15 | + | |
- | * May 6 -- 7. | + | |
- | You may in principle come at " | + | You may in principle come at " |