Wadim Zudilin

Ramanujan-type supercongruences☆

Abstract We present several supercongruences that may be viewed as p-adic analogues of Ramanujan-type series for 1 / π and 1 / π 2 , and prove three of these examples.

Irrationality of values of the Riemann zeta function

The paper deals with a generalization of Rivoals construction, which enables one to construct linear approximating forms in 1 and the values of the zeta function ζ(s) only at odd points. We prove theorems on the irrationality of the number ζ(s) for some odd integers s in a given segment of the set of positive integers. Using certain refined arithmetical estimates, we strengthen Rivoals original results on the linear independence of the ζ(s).

DENSITIES OF SHORT UNIFORM RANDOM WALKS

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

Generalizations of Clausen's Formula and algebraic transformations of Calabi–Yau differential equations

We provide certain unusual generalizations of Clausens and Orrs theorems for solutions of fourth-order and fifth-order generalized hypergeometric equations. As an application, we present several examples of algebraic transformations of Calabi-Yau differential equations.

From L -series of elliptic curves to Mahler measures

We prove the conjectural relations between Mahler measures and

NEW REPRESENTATIONS FOR APÉRY-LIKE SEQUENCES

L

Euler’s constant, q-logarithms, and formulas of Ramanujan and Gosper

-values of elliptic curves of conductors 20 and 24. We also present new hypergeometric expressions for

ON THE MAHLER MEASURE OF 1 + X + 1/X + Y + 1/Y

L

Dedekind's η-function and Rogers–Ramanujan identities

-values of CM elliptic curves of conductors 27 and 36. Furthermore, we prove a new functional equation for the Mahler measure of the polynomial family

New irrationality measures for q-logarithms

(1+X)(1+Y)(X+Y)-\alpha XY