Plan of the course Analytic and Cobinatorial Number Theory in summer term 2025/26

  • Finite fields and their combimatorial applications (Sidon sets)
  • Counting solutions of diagonal congruences mod p
  • An elementary proof of Weil's theorem on numbers of solutions of polynomial congruences mod p
  • Dirichlet's theorem on primes in AP
  • The PNT (analytic and elementary proofs)

    • lecture notes (preliminary, updated May 11, 2026). Exam questions: 1. The multiplicative group of a finite field F is cyclic and has exactly phi(|F|-1) generators (Theorem 1.6). 2. For every n we have Si(n) < n^{1/2} + n^{1/4} + 1 (Theorem 1.9). 3. The proposition on extensions of characters of groups (Proposition 3.4). 4. The 1st theorem of Mertens (Theorem 3.13). 5. Non-vanishing of L(1,chi) for real character chi (Theorem 3.21). 6. The instance of the Laplace transform needed in the proof of the PNT (Proposition 4.6).


  • Lecture 1, February 16, 2026 Finite fields


  • Lecture 2, February 23, 2026 Sidon sets


  • Lecture 3, March 2, 2026 Combinatorial applications of the Chevalley - Warning theorem


  • Lecture 4, March 9, 2026 A theorem from Borevich and Shafarevich on the number of solutions of a_1x^{r_1}+... +a_nx^{r_n} = 0 in F = Z_p (without proof). Beginning of the proof of Theorem 1: If N is the # of solutions of P(X, Y)=0 in Z_p, where P(X, Y) is abs. irreducible with d = deg(P) such that p > 250d^5, then |N - p|<(2d^5*p)^{1/2}.


  • Lecture 5, March 16, 2026 Continuation 1 of the proof of Theorem 1: If N is the # of solutions of P(X, Y)=0 in Z_p, where P(X, Y) is abs. irreducible with d = deg(P) such that p > 250d^5, then |N - p|<(2d^5*p)^{1/2}.


  • Lecture 6, March 23, 2026 Continuation 2 of the proof of Theorem 1: If N is the # of solutions of P(X, Y)=0 in Z_p, where P(X, Y) is abs. irreducible with d = deg(P) such that p > 250d^5, then |N - p|<(2d^5*p)^{1/2}.


  • Lecture 7, March 30, 2026 I planned to lecture on Dirichlet's theorem on primes in AP but nobody came.


  • April 6, 2026 no lecture - Easter Monday


  • Lecture 8, April 13, 2026 Shapiro's proof of Dirichlet's theorem on primes in AP.


  • Lecture 9, April 20, 2026 The sum L(chi) = sum_{n>0}chi(n)/n is not 0 for every non-principal Dirichlet character mod m in D(m).


  • Lecture 10, April 27, 2026 Two proofs that sum_{n>0} f(n)/n > 0 if f: N -> R is a completely multiplicative and strongly bounded function.


  • Lecture 11, May 4, 2026 Nobody came, but eventhough: The Prime Number Theorem


  • Lecture 12, May 11, 2026 I went through the content of the previous lecture


  • Lecture 13, May 18, 2026 Density Hales-Jewett theorem, overview