Equivalent formulations of the Riemann hypothesis based on the lines of constant phase

Result code in IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F18%3A00325860" target="_blank" >RIV/68407700:21340/18:00325860 - isvavai.cz</a>

Result on the web

<a href="http://iopscience.iop.org/article/10.1088/1402-4896/aabca9" target="_blank" >http://iopscience.iop.org/article/10.1088/1402-4896/aabca9</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1088/1402-4896/aabca9" target="_blank" >10.1088/1402-4896/aabca9</a>

Result language

angličtina

Original language name

Equivalent formulations of the Riemann hypothesis based on the 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: (a 1) f satisfies the functional equation f(1 - s) = f(s) for the complex argument s = σ + iτ, (a2) f is free of any pole, (a3) for large positive values of σ the phase θ of f increases in a monotonic way without a bound as τ 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 σ = 1/2, (R2) All lines of constant phase of f corresponding to $pm pi ,pm 2pi ,pm 3pi ,,...$ 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 π. 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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Type

J_imp - Article in a specialist periodical, which is included in the Web of Science database

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GA13-33906S" target="_blank" >GA13-33906S: Harnessing the Power of Quantum Walks</a><br>

Continuities

S - Specificky vyzkum na vysokych skolach

Publication year

2018

Confidentiality

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Name of the periodical

Physica Scripta

ISSN

0031-8949

e-ISSN

1402-4896

Volume of the periodical

93

Issue of the periodical within the volume

6

Country of publishing house

SE - SWEDEN

Number of pages

11

Pages from-to

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UT code for WoS article

000433131400001

EID of the result in the Scopus database

2-s2.0-85048118713