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Her vises forskjeller mellom den valgte versjonen og den nåværende versjonen av dokumentet.
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ma3150:2019v:start [2019-01-14] seip [Exercises] |
ma3150:2019v:start [2019-01-22] seip [Contents of the lectures] |
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[[larsmagnus.overlier@gmail.com|Lars Magnus Øverlier]] | [[larsmagnus.overlier@gmail.com|Lars Magnus Øverlier]] | ||
- | * First meeting January 14. | + | * {{ : |
===== Contents of the lectures ===== | ===== Contents of the lectures ===== | ||
* Lecture 1, January 8: The Poisson summation formula with the example of the Gaussian function. Definition of the Riemann zeta function \( \zeta(s) \). Euler product for \(\zeta(s)\) and Euler' | * Lecture 1, January 8: The Poisson summation formula with the example of the Gaussian function. Definition of the Riemann zeta function \( \zeta(s) \). Euler product for \(\zeta(s)\) and Euler' | ||
* Lecture 2, January 14: Apostol, sections 2.1 - 2.9. The Möbius function \(\mu(n)\), Euler' | * Lecture 2, January 14: Apostol, sections 2.1 - 2.9. The Möbius function \(\mu(n)\), Euler' | ||
- | * Lecture 3, January 15: Apostol, sections 2.10 - 2.12, 3.1 - 3.4. More on multiplicative functions, big oh notation, Abel summation and Euler' | + | * Lecture 3, January 15: Apostol, sections 2.10 - 2.12, 3.1 - 3.4 (see also Thm. 4.2 in 4.3 which implies (6) on page 54). More on multiplicative functions, big oh notation, Abel summation and the Euler--Maclaurin |
+ | * Lecture 4, January 21: Apostol, sections 3.5, 3.7, 3.11, 4.1 - 4.3. Further applications of Abel summation and the Euler--Maclaurin formula (more on the analytic continuation of \( \zeta(s) \), relation between \( \pi(x) \) and \( \psi(x) \), a weak version of Stirling' | ||
+ | * Lecture 5, January 22: Apostol 3.10 - 3.11, 4.5 - 4.8; see also Ch. 7 in Davenport. Chebyshev' | ||
+ | * **Special lecture** related to this course: Christian Skau, [[https:// | ||
+ | * Lecture 6, January 28: Preparation for our study of \(\zeta(s)\): | ||
+ | * Lecture 7, January 29: Riemann' | ||
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===== Exercises ===== | ===== Exercises ===== | ||
- | | + | You are welcome to work on the exercises in room 1329 SB2 on Friday 14: |
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===== Oral presentations ===== | ===== Oral presentations ===== | ||
- | During some of the last lectures of the course, 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' | ||
Linje 53: | Linje 62: | ||
- Ramanujan primes | - Ramanujan primes | ||
- General distribution of nontrivial zeros of \(\zeta(s)\) | - General distribution of nontrivial zeros of \(\zeta(s)\) | ||
- | - Zeros on the critical line, including density results | + | - Zeros on the critical line, including density results |
- 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 |