Elective subject Mathematics (Number theory)


TU Graz, SS 2020


Lecture Notes (still being edited):

Number theory SS20



Homeworks:

  1. Homework 1, due by April 22nd, 2020

  2. Homework 2, due by June 17th, 2020


Lessons:

  1. (4.03.2020): Intro, Arithmetic and multiplicative functions, Sum-function, Vinogradov's lemma and Möbius inversion formula

  2. (11.03.2020): Landau symbols, Direct estimates of some arithmetic functions, Abel summation formula

  3. (18.03.2020): Euler summation formula, Partial sums of \zeta(s), for s>1 and s=1, Euler-Mascheroni constant, Asymptotic for the median of Euler \phi-function, Density of k-free integers

  4. (25.03.2020): Dirichlet series, convergence half-plane, multiplication and uniqueness

  5. (1.04.2020): Dirichlet series of some popular arithmetic functions, Landau's theorem, Meromorphic continuation of the Riemann zeta-function for Re(s)>0 , Mellin transform of the Psi-function of Chebyshev

  6. (22.04.2020): Lower bounds of the prime-counting function, Chebyshev's theorem

  7. (29.04.2020): Merten's theorem, Structure of Newman's proof of the Prime Number Theorem, Analytic theorem of Newman for Laplace transforms and its corollary for Mellin transforms, from which we can deduce the PNT

  8. (6.05.2020): Finilizing the proof of the Prime Number Theorem, proof of the Analytic theorem of Newman

  9. (13.05.2020): Prime Number Theorem for Moebius and Liouville functions, characters of finite abelian groups, orthogonality

  10. (20.05.2020): Dirichlet characters, analytic properties of Dirichlet L-series

  11. (27.05.2020): The Prime number theorem for arithmetic progressions. Error terms in PNT and PNT for APs

  12. (3.06.2020): The Hardy-Littlewood circle method. Ternary Goldbach's problem and first estimates on the minor arcs

  13. (10.06.2020): Finilizing the estimate on the minor arcs, Proof of Claim 1 (Vinogradov)

  14. (17.06.2020): Ramanujan's sum, Statement of Siegel-Walfisz theorem and proving asymptotic formula for the integral over the major arcs

  15. (24.06.2020): Discussion of Homework 2 (notes of Gabriel Lipnik), Statement of other applications of the circle method (in binary Goldbach's problem and Waring's problem)