Interplay of multiplicative number theory and additive combinatorics

Project Information

MultNT

Grant agreement ID: 101058904

DOI

10.3030/101058904

Project terminated on 31 December 2024

EC signature date 4 July 2022

Start date 1 December 2022

End date 31 May 2025

Funded under

• Marie Skłodowska-Curie Actions (MSCA)

Total cost

No data

EU contribution

€ 215 534,40

Coordinated by

TURUN YLIOPISTO
Finland

Periodic Reporting for period 1 - MultNT (Interplay of multiplicative number theory and additive combinatorics)

Reporting period: 2022-12-01 to 2025-05-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

(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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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.

A plot of the summatory function of the Möbius function