equivalent statements

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… ►See Armitage (1989), Berry and Keating (1998, 1999), Keating (1993, 1999), and Sarnak (1999). …

… ►The prime number theorem (27.2.3) is equivalent to the statement … ►The Riemann hypothesis is equivalent to the statement …

… ►

\pi\left(x\right)-\operatorname{li}\left(x\right)

changes sign infinitely often as

x\to\infty

; see Littlewood (1914), Bays and Hudson (2000). ►The Riemann hypothesis (§25.10(i)) is equivalent to the statement that for every

x\geq 2657

, …

… ►The following three statements are equivalent:

w_{1}(z)

and

w_{2}(z)

comprise a fundamental pair in

D

;

\mathscr{W}\left\{w_{1}(z),w_{2}(z)\right\}

does not vanish in

D

;

w_{1}(z)

and

w_{2}(z)

are linearly independent, that is, the only constants

A

and

B

such that …