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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 [2019-01-14] seip [Oral presentations] |
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 ===== | ||
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- 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 |