Forskjeller

Her vises forskjeller mellom den valgte versjonen og den nåværende versjonen av dokumentet.

Lenk til denne sammenligningen

Begge sider forrige revisjon Forrige revisjon
Neste revisjon
Forrige revisjon
ma3150:2019v:start [2019-01-14]
seip [Oral presentations]
ma3150:2019v:start [2019-01-22] (nåværende versjon)
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'​s ​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
2019-01-14, Kristian Seip