DeMoivre theorem

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21: 23.23 Tables

… ►05, and in the case of $\mathrm{\wp }\left(z\right)$ the user may deduce values for complex $z$ by application of the addition theorem (23.10.1). …

22: 27.11 Asymptotic Formulas: Partial Sums

… ►where $\left(h,k\right)=1$, $k>0$. ►Letting $x\to \mathrm{\infty }$ in (27.11.9) or in (27.11.11) we see that there are infinitely many primes $p\equiv h(modk)$ if $h,k$ are coprime; this is Dirichlet’s theorem on primes in arithmetic progressions. … ►

27.11.15 $$\underset{x\to \mathrm{\infty }}{lim}\sum _{n\le x}\frac{\mu \left(n\right)\ln n}{n}=-1.$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.1 III

Referenced by:

§27.11, §27.11

Permalink:

http://dlmf.nist.gov/27.11.E15

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

►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 $p\equiv h(modk)$ is asymptotic to $x/(\varphi \left(k\right)\ln x)$ as $x\to \mathrm{\infty }$.

23: 27.12 Asymptotic Formulas: Primes

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Prime Number Theorem

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24: 1.10 Functions of a Complex Variable

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Picard’s Theorem

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§1.10(iv) Residue Theorem

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Rouché’s Theorem

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Lagrange Inversion Theorem

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Extended Inversion Theorem

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25: 1.4 Calculus of One Variable

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Mean Value Theorem

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Fundamental Theorem of Calculus

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First Mean Value Theorem

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Second Mean Value Theorem

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§1.4(vi) Taylor’s Theorem for Real Variables

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26: 35.2 Laplace Transform

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Convolution Theorem

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27: 24.10 Arithmetic Properties

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§24.10(i) Von Staudt–Clausen Theorem

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28: 1.6 Vectors and Vector-Valued Functions

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Green’s Theorem

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Stokes’s Theorem

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Gauss’s (or Divergence) Theorem

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Green’s Theorem (for Volume)

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29: 14.18 Sums

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§14.18(i) Expansion Theorem

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§14.18(ii) Addition Theorems

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30: 23.20 Mathematical Applications

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§23.20(ii) Elliptic Curves

… ►The geometric nature of this construction is illustrated in McKean and Moll (1999, §2.14), Koblitz (1993, §§6, 7), and Silverman and Tate (1992, Chapter 1, §§3, 4): each of these references makes a connection with the addition theorem (23.10.1). … ► $K$ always has the form $T\times {\mathbb{Z}}^r$ (Mordell’s Theorem: Silverman and Tate (1992, Chapter 3, §5)); the determination of $r$, the rank of $K$, raises questions of great difficulty, many of which are still open. …