Numero di Skewes

Nella teoria dei numeri, il termine numero di Skewes indica il più piccolo numero naturale x per il quale vale l'espressione

\pi (x)>\operatorname {Li} (x),

dove π (x) è la funzione enumerativa dei primi (cioè il numero di primi esistenti fino al numero x), e Li (x) è la funzione logaritmo integrale.

In pratica si tratta del più piccolo numero (che si è rivelato essere estremamente grande) per il quale π (x) risulta maggiore di Li (x).

L'esistenza di tale numero fu ipotizzata nel 1914 dal matematico John Littlewood, ma solo nel 1932 ne diede una dimostrazione. Littlewood provò anche che il segno della differenza π (x) – Li (x) cambia infinitamente spesso. Che tale numero esistesse non era affatto chiaro; infatti, l'evidenza numerica allora disponibile sembrava suggerire che π (x) fosse sempre minore di Li (x).^[1]

La dimostrazione di Littlewood, comunque, non fornì un esempio concreto del numero x; non era dunque un risultato costruttivo. Il matematico sudafricano Stanley Skewes, che era un allievo di Littlewood a Cambridge, nel 1933 dimostrò che, assumendo come vera l'ipotesi di Riemann, esiste un numero x per il quale π (x) > Li (x), inferiore a

e^{e^{e^{79}}}

(chiamato talvolta primo numero di Skewes), che è approssimativamente uguale a

10^{10^{8,85\times 10^{33}}}

^[2]

Nel 1955, senza l'assunzione che l'ipotesi di Riemann sia vera, Skewes dimostrò che deve esistere un valore di x inferiore a

10^{10^{10^{1000}}}

(chiamato talvolta secondo numero di Skewes).

Questi (enormi) estremi superiori sono stati in seguito ridotti considerevolmente. Senza assumere come vera l'ipotesi di Riemann, Herman te Riele nel 1987 trovò un estremo superiore di

7×10³⁷⁰

Una approssimazione migliore è 1,39822×10³¹⁶, scoperta da Bays e Hudson (2000). Il miglior valore per il primo attraversamento di zero è ora 1,397162914×10³¹⁶ (Demichel, 2005). Questo è, con un intervallo di confidenza molto elevato, il primo caso per cui si verifica π (x) > Li (x).

Note

^ Ancora oggi il valore più grande di Li (x) calcolato, per x = 10²⁴, è nettamente superiore al valore corrispondente di π (x).
^ Si tratta di un numero con ${10^{8,85\times 10^{33}}}$ cifre (vale a dire 8,85 milioni di miliardi di miliardi di miliardi di cifre). Un numero così immensamente grande è molto al di là della portata dei computer più potenti e avanzati. Volendo scrivere per esteso tale numero usando un comune blocco note a quadretti e inserendo una cifra per quadretto, si può calcolare che servirebbe un volume di carta di 1,78×10¹⁴ km³, pari al volume di un cubo con lato di circa 121000 km

Bibliografia

J.E. Littlewood, "Sur la distribution des nombres premiers", Comptes Rendus 158 (1914), pp. 1869-1872
S. Skewes, "On the difference π(x) – Li(x)", Journal of the London Mathematical Society 8 (1933), pp. 277-283
S. Skewes, "On the difference π(x) – Li(x) (II)", Proceedings of the London Mathematical Society 5 (1955), pp. 48–70
H.J.J. te Riele, "On the difference π(x) – Li(x)", Math. Comp. 48 (1987), pp. 323-328

Collegamenti esterni

(EN) Skewes Number, su daviddarling.info.

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