## 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