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ma3150:2019v:start [2019-01-14] seip [Oral presentations] |
ma3150:2019v:start [2019-03-19] seip [Guidance and consultation before the exam] |
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====== MA3150 Analytic Number Theory, Spring 2019 ====== | ====== MA3150 Analytic Number Theory, Spring 2019 ====== | ||
- | Analytic number theory studies the distribution of the prime numbers, based on methods from mathematical analysis. Of central importance is the study of the Riemann zeta function, which embodies both the additive and the multiplicative structure of the integers. It turns out that the localization | + | Analytic number theory studies the distribution of the prime numbers, based on methods from mathematical analysis. Of central importance is the study of the Riemann zeta function, which embodies both the additive and the multiplicative structure of the integers. It turns out that the location |
Key words for the course: Arithmetic and multiplicative functions, Abel summation and Möbius inversion, Dirichlet series and Euler products, the Riemann zeta function, the functional equation for the zeta function, the gamma function, The Mellin transformation and Perron' | Key words for the course: Arithmetic and multiplicative functions, Abel summation and Möbius inversion, Dirichlet series and Euler products, the Riemann zeta function, the functional equation for the zeta function, the gamma function, The Mellin transformation and Perron' | ||
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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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[[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: The gamma function continued; proof of Stirling' | ||
+ | * Lecture 8, February 4: Entire functions of order 1 and product representation of ξ(s) (see Ch. 11 and 12 in Davenport). (Notice our usage of Cauchy estimates and Jensen' | ||
+ | * Lecture 9, February 5: We have now two representations of ζ′(s)/ | ||
+ | * Lecture 10, February 11: We establish the argument principle and apply it to deduce the Riemann--von Mangoldt formula for \( N(T) \). We introduce and discuss the function \( S(T) \), which can be viewed as the argument of \( \zeta(s) \) on the critical line. We show that \( S(T)=O(\log T) \) (see Ch. 15 of Davenport). (The lecture is given by Kamalakshya Mahatab.) | ||
+ | * Lecture 11, February 12: We discuss the most interesting points of Exercise 1 and Exercise 2. (The lecture is given by Kamalakshya Mahatab.) | ||
+ | * **NB! The lecture on February 18 is cancelled** because of sickness absence of the lecturer. | ||
+ | * Lecture 12, February 19: We establish Perron' | ||
+ | * Lecture 13, February 22, 14:15-16, Room 734 SB2 (**This is an extra lecture**): We now apply Perron' | ||
+ | * Lecture 14, February 25: We establish the zero-free region of de la Vallée Poussin (Ch. 13 of Davenport) and use it to prove the prime number theorem (Ch. 18 of Davenport). | ||
+ | * Lecture 15, February 26: We start preparing for Dirichlet' | ||
+ | * Lecture 16, March 4. We start discussing Dirichlet' | ||
+ | * Lecture 17, March 5: We finish the proof of Dirichlet' | ||
+ | * Lecture 18, March 11. We will in the remaining lectures discuss the ideas that eventually lead to the prime number theorem for arithmetic progressions (see Ch. 20 and 22 in Davenport). We begin by introducing the notion of a primitive character, see Ch. 5 of Davenport and Ch. 8 of Apostol. We will mostly follow Apostol, relying on finite Fourier transforms (Gauss sums). This lecture will roughly cover 8.1 - 8.2, 8.5 - 8.7 in Apostol. (Be aware that Theorem 8.11 in Apostol is just Parseval' | ||
+ | * 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 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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===== Exercises ===== | ===== Exercises ===== | ||
- | | + | You are welcome to work on the exercises in room 1329 SB2 on Friday 14: |
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+ | * {{ : | ||
+ | * Exercise 3, from Apostol: 11.1 (a)--(d), 11.3, 13.2 (observing that \( A(x)=\pi(x)+\pi(x^{1/ | ||
+ | * Exercise 4, from Apostol: 6.14, 6.15, 7.1, 7.2, 7.3, 7.4, 7.6 plus {{ : | ||
+ | * Exercise 5, from Apostol: 8.5, 8.6, 8.7, 8.8, 8.9 (guidance offered on March 15 and March 22). | ||
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===== Oral presentations ===== | ===== Oral presentations ===== | ||
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- Mertens' | - Mertens' | ||
- | - The Bertrand--Chebyshev theorem, including Ramanujan and Erdős' | + | - The Bertrand--Chebyshev theorem, including Ramanujan and Erdős' |
- Ramanujan primes | - Ramanujan primes | ||
+ | - 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 | + | - 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 | ||
- 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 | + | - Elementary sieve methods and Brun's theorem on twin primes |
- 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. | + | 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. |
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+ | 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, | ||
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+ | ===== Exam, dates and location ===== | ||
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+ | 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. | ||
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+ | ===== Guidance and consultation before the exam ===== | ||
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+ | 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. | ||
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+ | You may in principle come at " | ||
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- | Each presentation should last for about 15--20 minutes. |