Periodic Reporting for period 1 - MultNT (Interplay of multiplicative number theory and additive combinatorics)
Période du rapport: 2022-12-01 au 2025-05-31
Many central problems in analytic number theory revolve around the distribution of prime numbers and the Möbius function, which is a function providing information about the prime factorisations of integers. The results of this project have given us new progress on some key problems on these sequences, particularly in the context of Gowers norms, which measure the pseudorandomness properties of a sequence and are an active topic of research. For the sequence of primes, the first quantitative bounds on its Gowers norms were obtained. For the Möbius function, the first results about its Gowers norms in all short intervals were achieved. These results pave the way for further progress on these questions.
(1) The main result concerning on Gowers uniformity of the Möbius function and other arithmetic sequences in short intervals has been published in Forum of Mathematics Pi, and further work on the topic is currently in preprint form.
(2) The main result concerning quantitative Gowers uniformity of the primes has been published in the Journal of the European Mathematical Society.
(3) The Hasse principle was proven for a large family of surfaces (generalised Châtelet surfaces). This article is currently in preprint form.
(4) Progress was made on a long-standing conjecture on sign changes of the Liouville function at polynomial arguments. This article was published in the American Journal of Mathematics.
(5) A new multiplicative transference principle was introduced and applied to make progress towards a conjecture of Erdős on products of primes in arithmetic progressions. This article has been published in Journal für die reine und angewandte Mathematik.
(6) The Erdős discrepancy problem over function fields was resolved in a paper published in Mathematische Annalen.
(2) The main result concerning quantitative Gowers uniformity of the primes has been published in the Journal of the European Mathematical Society.
(3) The Hasse principle was proven for a large family of surfaces (generalised Châtelet surfaces). This article is currently in preprint form.
(4) Progress was made on a long-standing conjecture on sign changes of the Liouville function at polynomial arguments. This article was published in the American Journal of Mathematics.
(5) A new multiplicative transference principle was introduced and applied to make progress towards a conjecture of Erdős on products of primes in arithmetic progressions. This article has been published in Journal für die reine und angewandte Mathematik.
(6) The Erdős discrepancy problem over function fields was resolved in a paper published in Mathematische Annalen.
The results on Gowers uniformity of the primes and the Möbius function obtained in this project have opened new avenues in the study of additive problems for these sequences. The results have been shown to have several applications in analytic number theory and ergodic theory in this project and in work of others. Several other results, such as the multiplicative transference principle developed, have also been found to have further applications.