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DIFFRACTION OF THE PRIMES AND OTHER SETS OF ZERO DENSITY

Part of: Optics, electromagnetic theory - General Elementary number theory

Published online by Cambridge University Press:  21 May 2025

ADAM HUMENIUK Open the ORCID record for ADAM HUMENIUK [Opens in a new window] ,
CHRISTOPHER RAMSEY Open the ORCID record for CHRISTOPHER RAMSEY [Opens in a new window]  and
NICOLAE STRUNGARU Open the ORCID record for NICOLAE STRUNGARU [Opens in a new window]
ADAM HUMENIUK
Affiliation:
Department of Mathematics and Computing, Mount Royal University, Calgary, Alberta, Canada e-mail: ahumeniuk@mtroyal.ca
CHRISTOPHER RAMSEY
Affiliation:
Department of Mathematics and Statistics, MacEwan University, Edmonton, Alberta, Canada e-mail: ramseyc5@macewan.ca
NICOLAE STRUNGARU*
Affiliation:
Department of Mathematics and Statistics, MacEwan University, Edmonton, Alberta, Canada, and Institute of Mathematics ‘Simon Stoilow’, Bucharest, Romania
*
e-mail: strungarun@macewan.ca
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Abstract

In this paper, we show that the diffraction of the primes is absolutely continuous, showing no bright spots (Bragg peaks). We introduce the notion of counting diffraction, extending the classical notion of (density) diffraction to sets of density zero. We develop the counting diffraction theory and give many examples of sets of zero density of all possible spectral types.

Keywords

diffractionautocorrelationprimes

MSC classification

Primary: 78A45: Diffraction, scattering
Secondary: 11A41: Primes
Type
Research Article
Information
Journal of the Australian Mathematical Society , First View , pp. 1 - 44
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc

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Footnotes

Communicated by Michael Coons

The first author was partially supported by the NSERC Discovery grant 2024-03883, the second author was supported by the NSERC Discovery grant 2019-05430, and the third author was supported by the NSERC Discovery grants 2020-00038 and 2024-04853.

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