Analytic Number Theory
Universität Wien, WS 14\15
Homeworks:
Homework 1, due October 24, 2014
Homework 2, due November 21, 2014
Solutions of Problem 3 about the sum of the divisor function over n^2+1
Homework 3, due January 9, 2015
Lessons:
Lesson 1 (3.10.2014): Intro, Arithmetic functions
Lesson 2 (10.10.2014): Direct estimates of multiplicative functions
Lesson 3 (17.10.2014): Abel transformation, Euler summation formula, application to the finite sum on 1/n^s
Lesson 4 (24.10.2014): Average estimates of arithmetic functions: Gauss circle problem, Dirichlet divisor problem, asymptotic formula for the average sum on the Euler function
Lesson 5 (31.10.2014): Dirichlet series: uniform convergence and abscissa of (absolute) convergence, multiplication of two Dirichlet series, representation as Euler product
Lesson 6 (7.11.2014): Connection between arithmetic functions and the Riemann zeta-function, Liouville and von Mangoldt functions. Analytic continuation of the Riemann zeta-function for Re(s)>0 .
Lesson 7 (14.11.2014): Chebyshev's Psi-function and its Mellin transform, density of square-free positive integers up to x. First not-so-trivial lower bounds on the prime-counting function, Chebyshev's theorem.
Lesson 8 (21.11.2014): Formulating an Analytic theorem of Newman and its Corollary A concerning analytic continuation of a certain Mellin transform. Deduction of the Prime Number Theorem from Corollary A.
Lesson 9 (28.11.2014): Proof of Mertens of the fact $\zeta(s)\neq 0$ for $Re(s)=1, s\neq 1$, proof of Corollary A of Newman. Other corollaries for the asymptotic behaviour of sums of arithmetic functions, given the holomorphicity properties of their Dirichlet series.
Lesson 10 (5.12.2014): Proof of the complex-analytic theorem of Newman. Discussion of Homework 2.
Lesson 11 (9.01.2015): Characters of finite abelian groups, orthogonality. Holomorphicity range of the Dirichlet L-series. L(\chi,s) is not zero on the line Re(s)=1 (method of Mertens and application of Landau's theorem in the case of real characters).
Lesson 12 (16.01.2015): The Prime number theorem for arithmetic progressions and Dirichlet's theorem for primes in arithmetic progressions. Discussion of Bombieri-Vinogradov theorem.
Lesson 13 (23.01.2015): Discussion of Dirichlet class number formula. Small gaps between primes I.
Lesson 14 (30.01.2015): Small gaps between primes II.