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ma3150:2019v:start [2019-01-14]
seip [Oral presentations]
ma3150:2019v:start [2019-01-22]
seip [Contents of the lectures]
Linje 34: Linje 34:
 [[larsmagnus.overlier@gmail.com|Lars Magnus Øverlier]] [[larsmagnus.overlier@gmail.com|Lars Magnus Øverlier]]
  
-  * First meeting January 14.+  * {{ :ma3150:2019v:report_refgroup_jan14-2019.pdf |Minutes}} from the 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's proof of the divergence of the series of reciprocals of the primes (from Davenport Ch. 1; se also Ch. Vanden Eynden, "Proofs that \(\sum 1/p\) diverges", Amer. Math. Monthly **87** (1980), 394--397). See {{ :ma3150:2019v:euler.pdf |this note}} for a more precise consequence of Euler's argument.   * 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's proof of the divergence of the series of reciprocals of the primes (from Davenport Ch. 1; se also Ch. Vanden Eynden, "Proofs that \(\sum 1/p\) diverges", Amer. Math. Monthly **87** (1980), 394--397). See {{ :ma3150:2019v:euler.pdf |this note}} for a more precise consequence of Euler's argument.
   * Lecture 2, January 14: Apostol, sections 2.1 - 2.9. The Möbius function \(\mu(n)\), Euler's totient function \(\varphi(n)\); basic properties and the relation between these two functions, Dirichlet convolution, Möbius inversion, the von Mangoldt function \(\Lambda(n) \), multiplicative functions.   * Lecture 2, January 14: Apostol, sections 2.1 - 2.9. The Möbius function \(\mu(n)\), Euler's totient function \(\varphi(n)\); basic properties and the relation between these two functions, Dirichlet convolution, Möbius inversion, the von Mangoldt function \(\Lambda(n) \), multiplicative functions.
-  * 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'summation formula, some asymptotic formulas. +  * 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 summation formula, some asymptotic formulas, meromorphic continuation of \( \zeta(s) \). (See {{ :ma3150:2019v:continuation_zeta.pdf |this note}} for both Abel summation and the Euler--Maclaurin formula applied to \( \zeta(s) \).) 
 +  * 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's formula), average order of \(d(n)\); for the average order of \(\varphi(n)\), see Exercise 1, problems 3.4, 3.5 in Apostol.  
 +  * Lecture 5, January 22: Apostol 3.10 - 3.11, 4.5 - 4.8; see also Ch. 7 in Davenport. Chebyshev's bounds on \( \pi(x) \) and Mertens's theorem on the asymptotics of \(\sum_{p\le x} p^{-1}\) (see {{ :ma3150:2019v:chebyshev_mertens.pdf |this note}} for a short route to these results; see also [[https://arxiv.org/pdf/math/0504289v3.pdf|here]] for an interesting account of Mertens's theorems). The sum \( \sum_{n\le x} \mu(n)/n \); the number of square-free numbers less than or equal to \( x \) (see Ex. 2.6 in Apostol for the "groundwork").  
 +  * **Special lecture** related to this course: Christian Skau, [[https://www.math.ntnu.no/seminarer/perler/|"A century of Brun's sieve - a momentous event in prime number history"]], January 25, 12:15 - 13:00, 1329 SB2. Coffee and cakes are served during the lecture. 
 +  * Lecture 6, January 28: Preparation for our study of \(\zeta(s)\): the basics of the Gamma function. See {{ :ma3150:2019v:gamma_function_notes.pdf |Note on the Gamma function}}. 
 +  * Lecture 7, January 29: Riemann's memoir and the functional equation for \(\zeta(s)\) (see Ch. 8 in Davenport). If time permits, we will then move on to Ch. 11 in Davenport, to arrive at the canonical factorization of \( \zeta(s) \) in terms of its pole and zeros. 
 + 
        
 ===== Exercises ===== ===== Exercises =====
  
-  * Exercise 1, from Apostol: 2.1, 2.2, 2.6, 2.21, 2.26, 3.1, 3.2, 3.4, 3.5, 4.7, 4.18, 4.19 (a).+You are welcome to work on the exercises in room 1329 SB2 on Friday 14:00--15:00, under my guidance, as indicated below. Solutions to the problems will be provided in due course. 
 + 
 +  * Exercise 1, from Apostol: 2.1, 2.2, 2.6, 2.21, 2.26, 3.1, 3.2, 3.4, 3.5, 4.7, 4.18, 4.19 (a) (guidance offered on January 18 and 25).
      
 ===== Oral presentations ===== ===== Oral presentations =====
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 (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
2023-02-16, Kristian Seip