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The research aims to explore the relationship between certain arithmetic sequences and oscillating matrix spectra derived from Fourier expansions.

June 4, 2026Open Access

Oscillating spectra associated to certain sequences draft1jun26

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Key Points

The research aims to explore the relationship between certain arithmetic sequences and oscillating matrix spectra derived from Fourier expansions.
Studied numerical sequences derived from Fourier expansions of cusp forms.
Observed and collected data on oscillatory behavior of associated matrices.
Initiated testing of vectors spaces formed by sequences exhibiting oscillatory behavior.
Identified oscillatory behavior in specific sequences within the study's observational range.
Proposed the existence of vector spaces formed by the sequences exhibiting oscillatory behavior.

Abstract

Identities originatingin the theory of symmetric functions expressmembers of arithmetic sequences as determinants. In this setting we studied numerically sequences \a (n) \₍ ₁derived from the Fourier expansions of cusp forms. We observed that, within the range of our observations, in certain cases the spectra of matrices we attached to these sequences exhibit oscillatory behavior. We collect data on this behavior in some other sequences. We propose that sequences exhibiting oscillatory behavior may form vector spaces. We initiate testing of this proposal.

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Barry Brent

University of Minnesota

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Oscillating spectra associated to certain sequences draft1jun26

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