## Analytic and Combinatorial Number Theory, NDMI045, summer term 2022/23

Seven exam questions. 1. Prove the Fundamental Theorem of Algebra. 2. Prove Liouville's inequality and obtain by it a specific transcendental number. 3. Prove that Euler's number e is transcendental. 4. Prove main properties of characters of a (finite Abelian) group. 5. Prove Selberg's identity (Ch. 3). 6. Prove Stirling's formula (Ch. 4). 7. Prove the functional equation for zeta (s) (Ch. 8). The questions 5 and 6 will not be examined on May 31 (as they are not yet written up in LN).
In this course we how one can use mathematical analysis in real and/or complex domain to count/handle/manipulate some algebraic/combinatorial/discrete/number-theoretic structures.
I plan to discuss these topics: Analytical proofs of irrationality and transcendence. Dirichlet's theorem on primes in aithmetic progression. The prime number theorem. Analytical Combinatorics. ...
lecture notes (updated on May 25)

Lecture 1 (February 17, 2023). Chapter 1. Algebraicity, irrationality and transcendence. The Fundamental Theorem of Algebra: every non-constant complex polynomial has a complex root.
Lecture 2 (February 24, 2023). Two proofs of Liouville's inequality. A corollary: the number sum_n 10^{-n!} is transcendental. Transcendence of Euler's number e.
Lecture 3 (March 3, 2023). Transcendence of pi.
Lecture 4 (March 10, 2023) Chapter 2. Dirichlet's theorem on primes in AP. The elementary proof of P. Erdos of particular cases of Dirichlet's theorem.
Lecture 5 (March 17, 2023) Conclusion of the elementary proof of P. Erdos of particular cases of Dirichlet's theorem. Analytical proof of Dirichlet's theorem - beginning
Lecture 6 (March 24, 2023) Analytical proof of Dirichlet's theorem - continuation
Lecture 7 (March 31, 2023) Analytical proof of Dirichlet's theorem - conclusion
(April 6, 2023) no lecture - Good Friday
Lecture 8 (April 14, 2023) Chapter 3. An elementary proof of PNT. I will lecture about this proof as given in the 5th edition of Hardy and Wright's An Introduction to the Theory of Numbers
Lecture 9 (April 21, 2023) Chapter 4. A proof of Stirlings asymptotic formula.
Lecture 10 (April 28, 2023) Chapter 5. Counting digraphs by multivariate Cauchy integral formula.
Lecture 9 (May 5, 2023) Chapter 6. A purely formal proof of Jacobi's four squares formula.
Lecture 10 (May 12, 2023) Chapter 7. The function zeta(s) and the Dirichlet series related to it. Chapter 1 in the book by E. C. Titchmarsh (and D. R. Heath-Brown) on the function zeta (s).
Lecture 11 (May 12, 2023) Chapter 8. The analytic character of zeta(s), and the functional equation. Chapter 2 ... .

May 2023