Equivalent formulations of the Riemann hypothesis based on lines of constant phase
Result description
We prove the equivalence of three formulations of the Riemann hypothesis for functions f defined by the four assumptions: (a1) f satisfies the functional equation f (1 - s) = f (s) for the complex argument s = sigma + i tau, (a2) f is free of any pole, (a3) for large positive values of s the phase. of f increases in a monotonic way without a bound as tau increases, and (a4) the zeros of f as well as of the first derivative f ' of f are simple zeros. The three equivalent formulations are: (R1) All zeros of f are located on the critical line sigma = 1/2, (R2) All lines of constant phase theta of f corresponding to +/-pi, +/- 2 pi, +/- 3 pi, ... merge with the critical line, and (R3) All points where f' vanishes are located on the critical line, and the phases of f at two consecutive zeros of f' differ by pi. Our proof relies on the topology of the lines of constant phase of f dictated by complex analysis and the assumptions (a1)-(a4). Moreover, we show that (R2) implies (R1) even in the absence of (a4). In this case (a4) is a consequence of (R2).
Keywords
continuous Newton methodlines of constant phaseRiemann hypothesis
The result's identifiers
Result code in IS VaVaI
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
Equivalent formulations of the Riemann hypothesis based on lines of constant phase
Original language description
We prove the equivalence of three formulations of the Riemann hypothesis for functions f defined by the four assumptions: (a1) f satisfies the functional equation f (1 - s) = f (s) for the complex argument s = sigma + i tau, (a2) f is free of any pole, (a3) for large positive values of s the phase. of f increases in a monotonic way without a bound as tau increases, and (a4) the zeros of f as well as of the first derivative f ' of f are simple zeros. The three equivalent formulations are: (R1) All zeros of f are located on the critical line sigma = 1/2, (R2) All lines of constant phase theta of f corresponding to +/-pi, +/- 2 pi, +/- 3 pi, ... merge with the critical line, and (R3) All points where f' vanishes are located on the critical line, and the phases of f at two consecutive zeros of f' differ by pi. Our proof relies on the topology of the lines of constant phase of f dictated by complex analysis and the assumptions (a1)-(a4). Moreover, we show that (R2) implies (R1) even in the absence of (a4). In this case (a4) is a consequence of (R2).
Czech name
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Czech description
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Classification
Type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Result continuities
Project
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Continuities
S - Specificky vyzkum na vysokych skolach
Others
Publication year
2018
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Physica Scripta
ISSN
0031-8949
e-ISSN
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Volume of the periodical
93
Issue of the periodical within the volume
6
Country of publishing house
GB - UNITED KINGDOM
Number of pages
11
Pages from-to
"065201(1)"-"065201(11)"
UT code for WoS article
000433131400001
EID of the result in the Scopus database
2-s2.0-85048118713
Basic information
Result type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
OECD FORD
Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)
Year of implementation
2018