# prime number theorem

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## 1—10 of 33 matching pages

##### 1: 27.12 Asymptotic Formulas: Primes

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###### Prime Number Theorem

…##### 2: 27.11 Asymptotic Formulas: Partial Sums

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►where $\left(h,k\right)=1$, $k>0$.
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27.11.15
$$\underset{x\to \mathrm{\infty}}{lim}\sum _{n\le x}\frac{\mu \left(n\right)\mathrm{ln}n}{n}=-1.$$

►Each of (27.11.13)–(27.11.15) is equivalent to the prime number theorem (27.2.3).
*The prime number theorem for arithmetic progressions*—an extension of (27.2.3) and first proved in de la Vallée Poussin (1896a, b)—states that if $\left(h,k\right)=1$, then the number of primes $p\le x$ with $p\equiv h\phantom{\rule{0.949em}{0ex}}(modk)$ is asymptotic to $x/(\varphi \left(k\right)\mathrm{ln}x)$ as $x\to \mathrm{\infty}$.##### 3: 27.2 Functions

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►(See Gauss (1863, Band II, pp. 437–477) and Legendre (1808, p. 394).)
►This result, first proved in Hadamard (1896) and de la Vallée Poussin (1896a, b), is known as the

*prime number theorem*. …This is the number of positive integers $\le n$ that are relatively prime to $n$; $\varphi \left(n\right)$ is*Euler’s totient*. …##### 4: Tom M. Apostol

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►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem).
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##### 5: 25.16 Mathematical Applications

##### 6: 25.10 Zeros

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►Also, $\zeta \left(s\right)\ne 0$ for $\mathrm{\Re}s=1$, a property first established in Hadamard (1896) and de la Vallée Poussin (1896a, b) in the proof of the prime number theorem (25.16.3).
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##### 7: Bibliography

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A Centennial History of the Prime Number Theorem.
In Number Theory,
Trends Math., pp. 1–14.
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##### 8: 25.15 Dirichlet $L$-functions

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►Related results are:
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##### 9: Errata

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Paragraph Prime Number Theorem (in §27.12)
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The largest known prime, which is a Mersenne prime, was updated from ${2}^{43,112,609}-1$ (2009) to ${2}^{82,589,933}-1$ (2018).