WAMS research school
Topics in Analytic and Transcendental
Number Theory

Institute for Advanced Studies in Basic Sciences (IASBS)
Zanjan, Iran - July 1th - July 13th 2017

Home Committees Courses Notes Schedule
Elementary methods in analytic number theory
Lecturers: Mehdi Hassani & Jean-Marc Deshouillers
  • Part I. Elementary and combinatorial number theory
    • Generalities on primes
    • Arithmetical functions
    • Dirichlet convolution
    • Formal Euler products
    • Mobius inversion formulas
    • Chebyshev's upper and lower bounds for primes up to X.
    • Brun's sieve and twin primes
  • Part II. Calculus
    • Mean value of some arithmetical functions
    • Divisors and the hyperbola
    • Sums of two squares and the circle problem
  • Part III. Fourier
    • Upper bounds for trigonometric sums : van der Corput
    • Application to Voronoi's theorem
    • Application to the circle problem
  • Part IV. Heuristics and probabilistic methods (Time permitting)
    • Normal orders of arithmetic functions
    • Distribution modulo 1
  • Gerald Tenenbaum Course on analytic and probabilistic number theory (published in English by AMS) is as a very good reference for the above course and related topics.
Analytic Problems for Elliptic curves

Lecturers: Amir Akbary & Francesco Pappalardi
  • Part I : Algebraic Number Theory
    • basics of algebraic number theory (Number fields, rings of integers, splitting of primes)
    • splitting of primes in cyclotomic fields and primes in arithmetic progressions
  • Part II : Elliptic Curves
    • basics of elliptic curves (structure of the group of rational points, elliptic curves over finite fields)
    • reduction mod p of elliptic curves
    • splitting of primes in division fields of elliptic curves
  • Part III : Analytic Results
    • Prime number theorem for arithmetic progressions
    • Bombieri-Vinogradov theorem
    • Chebotarev density theorem
  • Part IV : Applications
    • classical Titchmarch divisor problem
    • elliptic Titchmarsh divisor problem
    • Serre's cyclicity problem
    • Artin's conjecture and its elliptic analogue (time permitting)
  • Lawrence C. Washington, Elliptic Curves: Number Theory and Crptography. Chapman & Hall (CRC) 2003. Joseph
  • H. Silverman, The Arithmetic of Elliptic Curves. Springer GTM 2009
  • Alina Carmen Cojocaru, Questions about the reductions modulo primes of an elliptic curve. Number theory, 61-79, CRM Proc. Lecture Notes, 36, Amer. Math. Soc., Providence, RI, 2004