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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F61989100%3A27240%2F18%3A10239065" target="_blank" >RIV/61989100:27240/18:10239065 - 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 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: (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 &apos; 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&apos; vanishes are located on the critical line, and the phases of f at two consecutive zeros of f&apos; 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).

Název v anglickém jazyce

Equivalent formulations of the Riemann hypothesis based on 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: (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 &apos; 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&apos; vanishes are located on the critical line, and the phases of f at two consecutive zeros of f&apos; 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).

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

—

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

—

Svazek periodika

93

Číslo periodika v rámci svazku

6

Stát vydavatele periodika

GB - Spojené království Velké Británie a Severního Irska

Počet stran výsledku

11

Strana od-do

"065201(1)"-"065201(11)"

Kód UT WoS článku

000433131400001

EID výsledku v databázi Scopus

2-s2.0-85048118713