Riemannova domneva: Razlika med redakcijama

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m+/dp
(m+/dp)
gosta v [[Hilbertov prostor|Hilbertovem prostoru]] ''L''<sup>2</sup>(0,1) kvadratnointegrabilnih funkcij v [[enotski interval|enotskem intervalu]]. [[Arne Beurling|Beurling]]<ref>{{sktxt|Beurling|1955}}.</ref> je leta 1955 to razširil in pokazal, da funkcija ζ nima ničel z realnim delom večjim od 1/''p'', če in samo če je ta funkcijski prostor gost v ''L<sup>p</sup>''(0,1).
 
[[Weilov kriterij]] je izjava, da je pozitivnost določene funkcije enakovredna Riemannovi domnevi. [[Enrico Bombieri|Bombieri]] in [[Jeffrey Lagarias|Lagarias]] sta pokazala, da sorodni [[Lijev kriterij|Li-Keiperjev kriterij]] sledi iz Weilovega kriterija za [[posplošena Riemannova domneva|posplošeno Riemannovo domnevo]]. Li-Keiperjev kriterij je izjava o pozitivnosti določenega zaporedja števil, ki je enakovredna Riemannovi domnevi.<ref>{{sktxt|Li|1997}}.</ref> Vse vrednosti členov [[zaporedje|zaporedja]] λ<sub>''n''</sub>&nbsp;&gt;&nbsp;0 za vse pozitivne ''n''. Števila <math>\lambda_n\, </math>, imenovana Li-Keiperjeve konstante<ref name="coffey_2007">{{sktxt|Coffey|2007}}.</ref> ali Keiper-Lijevi koeficienti,<ref>{{sktxt|Arias de Reyna|2011}}.</ref> se lahko izrazijo z netrivialnimi ničlami Riemannove funkcije ζ:
 
: <math> P_{n} (\gamma_{0}, \ldots , \gamma_{n-1}) = \lambda_n = \sum_{\rho} \left[1- \left(1-\frac{1}{\rho}\right)^n\right] \!\, , </math>
 
kjer vsota poteka po ρ, netrivialnih ničlah Riemannove funkcije ζ po vrsti <math>|\Im(\rho)|</math>, <math>P_{n}\, </math> pa je [[polinom]] končno mnogo [[Stieltjesove konstante|Steiltjesovih konstant]].<ref name="matijasevič_2014">{{sktxt|Matijasevič|2014}}.</ref> To [[pogojna konvergenca|pogojno konvergentno]] vsoto je treba razumeti v smislu, ki se po navadi rabi v [[teorija števil|teoriji števil]], tako da velja limita:
 
: <math> \sum_{\rho} = \lim_{N\to\infty} \sum_{|\Im(\rho)|\le N} \rho \!\, . </math>
 
[[Andreas Speiser|Speiser]]<ref>{{sktxt|Speiser|1934}}.</ref> je leta 1934 dokazal, da je Riemannova domneva enakovredna izjavi, da odvod funkcije ζ(''s''):
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