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# Fourier Methods and Multiplicative Analysis

**Alternative title:** Fourier-metoder og multiplikativ analyse

#### Awarded: NOK 12.0 mill.

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Project Number:

334466

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Project Period:

2023 - 2028

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This project centers around the Riemann zeta function which carries information about the distribution of the prime numbers, i.e., the positive integers 2, 3, 5, 7, 11 etc. that are only divisible by 1 and themselves. Such numbers play a crucial role in secure public-key encryption systems. The famous Riemann hypothesis predicts the location of the so-called nontrivial zeros of the zeta function. If true, the Riemann hypothesis would give a precise asymptotic formula for the distribution of the prime numbers.
The project explores links between the nontrivial zeros of the zeta function and the uncertainty principle of physics and signal analysis. It also investigates how the connection between the prime numbers and the nontrivial zeros of the zeta function can be understood in terms of quasicrystals, which are structures that are ordered but not periodic.
The study of the distribution of the prime numbers is largely a matter of understanding the interplay between additive and multiplicative structures. The project develops a branch of mathematical analysis, called multiplicative analysis, that studies this kind of interplay from a functional analytic point of view.

The proposal consists of three topics: (A) Analysis on the Riemann zeta function, (B) multiplicative analysis, and (C) Fourier interpolation and quasicrystals. Parts (B) and (C) represent emerging fields of analysis and are interconnected via their link to part (A). The proposed research aims at shaping the new fields (B) and (C) and at the same time advancing related analytic methods pertaining to part (A).
Part (A) aims at developing analytic techniques and viewpoints that are of specific interest to the Riemann zeta function and other L-functions. The project will in particular pursue possible number theoretic applications of Fourier interpolation and quasicrystals in this context.
Part (B) will develop the theory of Hardy spaces of Dirichlet series and specifically study composition operators acting on such spaces and spectral resolution of Mellin type operators. The project aims at solving embedding problems, with special emphasis on clarifying the scope of contractive embeddings and describing suitable classes of Carleson measures. An additional goal of part (B) is to describe curves and measures for which Fourier restriction is admitted.
Part (C) will develop the general theory of Fourier interpolation and quasicrystals. The proposed research aims at clarifying the scope of Fourier interpolation and at unifying this emerging theory with classical time–frequency analysis. This involves obtaining a better understanding of the link between Fourier interpolation and crystalline measures, as well as determining whether the full structure inherent in a quasicrystal is indeed needed for universal sampling.
The project will support 9 man-years of research carried out by three PhD students.

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1.6BILL. NOKtotal funding in the programme period758PROJECTShave received funding in the programme period1SOURCEhas financed the programme