Operator-valued zeta functions and Fourier analysis Article Swipe
Carl M. Bender
,
Dorje C. Brody
·
YOU?
·
· 2019
· Open Access
·
· DOI: https://doi.org/10.1088/1751-8121/ab25fa
YOU?
·
· 2019
· Open Access
·
· DOI: https://doi.org/10.1088/1751-8121/ab25fa
The Riemann zeta function Ϛ(s) is defined as the infinite sum \n∑∞n=1n-s, which \nconverges when Re s ˃ 1. The Riemann hypothesis asserts that the nontrivial zeros \nof Ϛ(s) lie on the line Re s = ½ . Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex s for which the defining sum does not \nconverge. This analytic continuation is ordinarily performed by using a functional \nequation. In this paper it is argued that one can investigate some properties of \nthe Riemann zeta function in the region Re s ˂ 1 by allowing operator-valued zeta \nfunctions to act on test functions. As an illustration, it is shown that the locations \nof the trivial zeros can be determined purely from a Fourier series, without relying \non an explicit analytic continuation of the functional equation satisfied by Ϛ(s).
Related Topics
Concepts
Riemann zeta function
Analytic continuation
Riemann hypothesis
Mathematics
Riemann Xi function
Continuation
Operator (biology)
Fourier series
Arithmetic zeta function
Mathematical analysis
Fourier analysis
Functional equation
Explicit formulae
Zeta function regularization
Analytic function
Pure mathematics
Prime zeta function
Fourier transform
Differential equation
Computer science
Gene
Chemistry
Biochemistry
Repressor
Programming language
Transcription factor
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.1088/1751-8121/ab25fa
- OA Status
- green
- Cited By
- 7
- References
- 9
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W2894656563
All OpenAlex metadata
Raw OpenAlex JSON
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https://openalex.org/W2894656563Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.1088/1751-8121/ab25faDigital Object Identifier
- Title
-
Operator-valued zeta functions and Fourier analysisWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2019Year of publication
- Publication date
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2019-05-31Full publication date if available
- Authors
-
Carl M. Bender, Dorje C. BrodyList of authors in order
- Landing page
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https://doi.org/10.1088/1751-8121/ab25faPublisher landing page
- Open access
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YesWhether a free full text is available
- OA status
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greenOpen access status per OpenAlex
- OA URL
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https://arxiv.org/pdf/1810.01821Direct OA link when available
- Concepts
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Riemann zeta function, Analytic continuation, Riemann hypothesis, Mathematics, Riemann Xi function, Continuation, Operator (biology), Fourier series, Arithmetic zeta function, Mathematical analysis, Fourier analysis, Functional equation, Explicit formulae, Zeta function regularization, Analytic function, Pure mathematics, Prime zeta function, Fourier transform, Differential equation, Computer science, Gene, Chemistry, Biochemistry, Repressor, Programming language, Transcription factorTop concepts (fields/topics) attached by OpenAlex
- Cited by
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7Total citation count in OpenAlex
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2025: 2, 2023: 2, 2022: 1, 2021: 2Per-year citation counts (last 5 years)
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10Other works algorithmically related by OpenAlex
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