TN510 - Introduction to Analytic Number Theory

2017/2018 - 2nd Semester - Credits 6.

Gneral Information

Teacher Francesco Pappalardi
Office hours TBA
Office 209
Telephone 06 57338243
E-mail pappa at mat dot uniroma3 dot it
Classes:
Tuesday: 14-16 (Classroom C)
Thursday: 14-16 (Classroom C)
Friday: 14-16 (Classroom 211)
COURSE DESCRIPTION

IMPORTANT INFORMATION:
  1. (Thesday 28 February) Introduction of the course. The notion of Dirichlet Character
  2. (Tuesday 6 March) Constructions and examples of Dirichlet Characters
  3. (Thursday 8 March) L-functions, the Riemann zeta function, Euler product, continuation to σ>0,
  4. (Friday 9 March) Proof of the Dirichlet Theorem for primes in arithmetic progressions (first part), linear combination of logarithm of L-functions and ortogonality relations.
  5. (Tuesday 13 March) Gauss sums. Last step of the Proof of Dirichlet Theorem for primes in arithmetic progressions. The Dirichlet Hyperbola method.
  6. (Thursday 15 March) Introduction to Complex Analysis: the notion of holomorhic function
  7. (Friday 16 March) Introduction to Complex Analysis: power series of holomorphic functions
  8. (Tuesday 20 March) Introduction to Complex Analysis: complex integration
  9. (Friday 23 March) Exercises of complex analysis (Andam Mustafa)
  10. (Tuesday 27 March) Application of complex analysis to the proof that L(1,χ)≠0 when χ is a real Dirichlet Character, The Riemann memoire.
  11. (Thursday 29 March) Estensione meromorfa per ζ, Gli zeri banali per ζ e simmetria degli zeri.
  12. (Thursday 5 April) [Filippo Tolli] entire functions of finite order, characterization of entire functions of finite order without zeros. Jensen's Formula and consequences.
  13. (Friday 6 April) [Filippo Tolli] Weierstrass product in the case when \sum 1/|zn| (where zn are the zeroes) is convergent. Hadamard factorization Theorem for entire funztions of order 1. Example of the factorization of sin(πz). Exercises and notes.
  14. (Tuesday 10 April) The function ξ is entire of order 1. The infinite product for ξ. B = (log(4π)-γ-2)/2.
  15. (Thursday 12 April) Sum of reciprocals of the zeros. The zeroes ρ= β+iγ satisfy |γ|>6.
  16. (Friday 13 April) the ζ(1+it)≠0 for all t. The zero free region (beginning)
  17. (Tuesday 17 April) The zero free region (end)
  18. (Thursday 26 April) The Explicit formulas for the Riemann zeta function
  19. (Friday 27 April) The analytic proof of the prime number Theorem.
  20. (Thursday 3 May) The analytic proof of the prime number Theorem (continuation). The prime number Theorem for arithmetic progressions.
  21. (Friday 4 May) The prime number Theorem for arithmetic progressions (continuation).
  22. (Tuesday 8 May) [Filippo Tolli] materiale di preparazione alle lezioni della settimana
  23. (Thursday 10 May) [Filippo Tolli]
  24. (Friday 11 May) [Filippo Tolli]
  25. (Tuesday 15 May) seminars by Andam Ali Mustafa and Manoj Gyawali

SCHEDULE OF STUDENTS' SEMINARS

  1. (15/5/18 ore 14-15) the Method of van der Corput. Andam Ali Mustafa
  2. (15/5/18 ore 15-16) the Gamma function. Manoj Gyawali
  3. (4/6/18 ore 11-12) evaluating ζ(2) I. Federica Catalano
  4. (4/6/18 ore 12-13) evaluating ζ(2) II. Marco Tabarrini
  5. (5/6/18 ore 11-12) Smooth Numbers I. Alessio Rampogni
  6. (5/6/18 ore 12-13) Smooth Numbers II. Chiara Melillo
  7. (5/6/18 ore 14-15) Smooth Numbers III. Giulia Ciccone
  8. (6/6/18 ore 11-12) Chebicev and Merten's Theorem for primes and Merten's Theorem for primes in arithmetic progressions I. Alessadra Amerini
  9. (6/6/18 ore 12-13) Chebicev and Merten's Theorem for primes and Merten's Theorem for primes in arithmetic progressions II. Chiara Camerini
  10. (7/6/18 ore 11-12) Vertical distribution of zeros and density estimates I. Federico Antonelli
  11. (7/6/18 ore 12-13) Vertical distribution of zeros and density estimates II. Marco Parsi
  12. (7/6/18 ore 12-13) Vertical distribution of zeros and density estimates III. Pierpaolo Colagè
  13. (8/6/18 ore 11-12) Dirichlet class number formula I. Francesca Bernieri
  14. (8/6/18 ore 12-13) Dirichlet class number formula II. Claudia del Chiaro
  15. (8/6/18 ore 14-15) Dirichlet class number formula III. Michela Ferrante
  16. (8/6/18 ore 14-15) The circle method by Hardy. Silvia De Francisci

COURSE OUTLINE

Literature:


Suggested Books

  • Harold Davenport. Multiplicative Number Theory. (Graduate Texts in Mathematics) Springer.
  • Gerald Tenenbaum. Introduction to Analytic and Probabilistic Number Theory. (Cambridge Studies in Advanced Mathematics) CUP.
  • Tom M. Apostol. Introduction to Analytic Number Theory. (Undergraduate Texts in Mathematics) Springer.

    Other Text

  • Henryk Iwaniec, Emmanuel Kowalski. Analytic Number Theory (Colloquium Publications, Vol. 53) (Colloquium Publications (Amer Mathematical Soc)).
  • Paul T. Bateman, Harold G. Diamond. Analytic Number Theory: An Introductory Course