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

Kód výsledku v 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>

Výsledek na webu

<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>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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).

Název v anglickém jazyce

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

Popis výsledku anglicky

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).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GA13-33906S" target="_blank" >GA13-33906S: Využití potenciálu kvantových procházek</a><br>

Návaznosti

S - Specificky vyzkum na vysokych skolach

Rok uplatnění

2018

Kód důvěrnosti údajů

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

Název periodika

Physica Scripta

ISSN

0031-8949

e-ISSN

1402-4896

Svazek periodika

93

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

SE - Švédské království

Počet stran výsledku

11

Strana od-do

—

Kód UT WoS článku

000433131400001

EID výsledku v databázi Scopus

2-s2.0-85048118713