# equivalent statements

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4 matching pages ♦

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## 4 matching pages

##### 1: 25.17 Physical Applications

##### 2: 25.16 Mathematical Applications

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►The prime number theorem (27.2.3) is equivalent to the statement
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►The Riemann hypothesis is equivalent to the statement
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##### 3: 27.12 Asymptotic Formulas: Primes

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$\pi \left(x\right)-\mathrm{li}\left(x\right)$ changes sign infinitely often as $x\to \mathrm{\infty}$; see Littlewood (1914), Bays and Hudson (2000).
►The

*Riemann hypothesis*(§25.10(i)) is equivalent to the statement that for every $x\ge 2657$, …##### 4: 1.13 Differential Equations

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►The following three statements are equivalent: ${w}_{1}(z)$ and ${w}_{2}(z)$ comprise a fundamental pair in $D$; $\mathcal{W}\left\{{w}_{1}(z),{w}_{2}(z)\right\}$ 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 …