Arithmetic Randomness

The aim of the proposed Austrian-French joint project "Arithmetic Randomness" (Aléa arithmétique, Arithmetische Zufälligkeit) is to make progress on various important open conjectures and questions in analytic, metric, probabilistic and additive number theory. All problems address random-like phenomena in arithmetic problems. The main areas of interest are distributional properties of prime numbers, its relations to number systems and Diophantine problems. As modern cryptographic systems are heavily based on prime numbers and pseudorandom numbers, it is important to understand the distribution of the underlying arithmetic quantities and to quantify its relation to randomness. One of the most striking open conjectures in this area is Sarnak`s conjecture which currently attracts considerable interest. It states that the Möbius function is "orthogonal" i.e. is very different from any sequence that is realized in a deterministic dynamical system. The Möbius function is an important sequenc e that carries essential information about prime numbers: It`s values mu(n) are 1,-1, or 0, depending on whether n has an even number of prime factors, an odd number, or whether a prime factor occurs more than once. There have been many important partial results, including by project member Müllner for automatic sequences which, informally speaking, are sequences that relate the value of its terms to the digits of its base-q representation. There are a number of further related important open conjectures we intend to work on, including the Duffin-Schaeffer conjecture. This project is a cooperation of researchers at TU Graz and TU Vienna, with colleages in Nancy, Marseille, Bordeaux, St. Etienne and Calais, under the direction of