## “Bulletin Board”

School of Mathematics - January 2, 2005

### Mathematical Lectures and Short Course

#### Wadim Zudilin from Moscow Lomonsov State University is going to visit School of Mathematics of IPM during January 17-February 6, 2005. During his visit, Prof. Zudilin will present two lectures and will conduct a short course in "Special Functions in Number Theory" at IPM.

Wadim Zudilin from Moscow Lomonsov State University is going to visit School of Mathematics of IPM during January 17-February 6, 2005. During his visit, Prof. Zudilin will present two lectures and will conduct a short course in "Special Functions in Number Theory" at IPM.

Short Course

Special Functions in Number Theory
 1. Hypergeometric Series Jan. 22, 10:00-12:00 Gauss hypergeometric series. Integral representations. Gauss' summation formula. Irrationality of log(2). Generalized hypergeometric series. Summation and transformation formulae. 2. Hypergeometric Differential Equation Jan. 23, 10:00-12:00 Hypergeometric equation. Monodromy representation. Solutions around singular points. Schwarz map and Schwarz triangles. Schwarz's reflection principle. 3. Automorphic and Modular Functions Jan. 30, 10:00-12:00 Non-linear differential equations. Modular group. Modular forms. Eisenstein series and Ramanujan's system of differential equations. Modular invariant. Modular polynomial. Algebraic independence results for modular functions. 4. Ellptic Functions Feb. 1, 10:00-12:00 Lattices in $\mathbb C$ and elliptic curves. Isomorphism classes of elliptic curves. Elliptic functions, basic properties. Weierstrass $\wp$-function. Laurent expansion of the $\wp$-function. Algebraic differential equation for the $\wp$-function. Cubic plane curves. Realization of elliptic curves in terms of the $\wp$-function. Applications in number theory.

Lectures

Computing Mathematical Constants
Jan. 24, 10:00-12:00
Abstract. Mathematical constants, like $\pi$, Euler's constant $\gamma$, Catalan's constant $G$, and values of the Riemann zeta function at positive integers, are a curious part of mathematics. We present two major (pure mathematical!) methods in the high-precision evaluation of such constants, namely, (1) accelerated convergence series, and (2) polynomial recurrences. We also explain interaction of the methods with the study of arithmetic properties of the constants and indicate some open problems in this respect.
Hypergeometric Integrals and Irrationality Proofs
Jan. 25, 10:00-12:00
Abstract. One of the hard problems in number theory is proving that values of the Riemann zeta function
$$\zeta(s)=\sum_{n=1}^\infty\frac1{n^s}$$
at odd integers $s\ge3$ are irrational. Several attempts were made to generalize Ap\'ery's remarkable proof (1978) of the irrationality of $\zeta(3)$, and to the moment we dispose of some new partial arithmetic results for \$\zeta(5),\zeta(7),\zeta(9), .... The aim of the talk is to show that hypergeometric integrals appear quite naturally in the corresponding irrationality proofs.