Arithmetic Randomness – ArithRand
The aim of the proposed Austrian-French joint project "Arithmetic Randomness" (Aléa arithmétique, Zahlen und Zufall) 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 problems in Diophantine approximation. 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 to any sequence that is realized in a deterministic dynamical system. While still open in its full generality, considerable progress has been obtained with respect to special classes of sequences. We mention the solution of the conjecture by Mu¨llner (2017) for automatic sequences which, informally speaking, are sequences that relate the value of its terms to the digits of its base-q representation. Besides other tools, the work is based on the method of Mauduit and Rivat (2010) on the digits of primes. Sarnak's conjecture is closely related to Chowla's conjecture that in turn is the problem to study correlations of the Möbius function. In the last years, various breakthroughs have been obtained in these areas. We mention the results by Green and Tao (2012) concerning the orthogonality of nilsequences, by Bourgain, Sarnak and Ziegler (2013) concerning the disjointness of the horocycle flow, by Frantzikinakis and Host (2018) on the logarithm Sarnak conjecture for zero entropy topological systems with only countably many ergodic measures, by Green and Bourgain (2012-16) concerning the computational complexity of the Möbius function, by Bourgain (2017) on the number of prime numbers with preassigned digits, by Matomäki and Radziwill (2016) on multiplicative functions in short intervals, by Tao (2016) on a logarithmic version of the Chowla conjecture, and by Maynard (2019) about prime numbers with restricted digits. The proofs rely on tools of analytic number theory, geometry of numbers, Diophantine considerations as well as dynamical systems. Probabilistic and statistical aspects of Diophantine approximation play an important role in the study of the pair correlation of sequences, which are generally wide out of reach for deterministic sequences. Several new tools are available in the metric setting and, somewhat surprisingly, these problems have a close connection with the value distribution of the Riemann zeta function in the critical strip.
Monsieur Thomas Stoll (Institut Elie Cartan de Lorraine)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
IMB Institut de mathématiques de Bordeaux
Technische Universität Wien / Institute of Discrete Mathematics and Geometry
I2M Institut de Mathématiques de Marseille
EA2597 LABORATOIRE DE MATHEMATIQUES PURES ET APPLIQUEES JOSEPH LIOUVILLE
ICJ Institut Camille Jordan
Technische Universität Graz / Institut für Analysis und Zahlentheorie
IECL Institut Elie Cartan de Lorraine
Help of the ANR 371,909 euros
Beginning and duration of the scientific project: January 2021 - 48 Months