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1: 27.20 Methods of Computation: Other Number-Theoretic Functions

§27.20 Methods of Computation: Other Number-Theoretic Functions

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2: 27.17 Other Applications

§27.17 Other Applications

►Reed et al. (1990, pp. 458–470) describes a number-theoretic approach to Fourier analysis (called the arithmetic Fourier transform) that uses the Möbius inversion (27.5.7) to increase efficiency in computing coefficients of Fourier series. …

3: 6.16 Mathematical Applications

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§6.16(ii) Number-Theoretic Significance of $\mathrm{li}\left(x\right)$

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Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

4: 27.5 Inversion Formulas

§27.5 Inversion Formulas

… ►The set of all number-theoretic functions $f$ with $f(1)\ne 0$ forms an abelian group under Dirichlet multiplication, with the function $\lfloor 1/n\rfloor$ in (27.2.5) as identity element; see Apostol (1976, p. 129). …Generating functions yield many relations connecting number-theoretic functions. … ►

27.5.2 $$\sum _{d|n}\mu \left(d\right)=\lfloor \frac{1}{n}\rfloor ,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\lfloor x\rfloor$: floor of $x$ , $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.1 II.B (in slightly different form)

Referenced by:

§27.5

Permalink:

http://dlmf.nist.gov/27.5.E2

Encodings:

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See also:

Annotations for §27.5 and Ch.27

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5: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

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27.8.4 $$\chi \left(n\right)=0,$$ $\left(n,k\right)>1$.

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\left(m,n\right)$: greatest common divisor (gcd), $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E4

Encodings:

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See also:

Annotations for §27.8 and Ch.27

… ►If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation ►

27.8.6

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $\varphi \left(n\right)$: Euler’s totient, $\overline{z}$: complex conjugate, $\equiv$: modular equivalence, $k$: positive integer, $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.8.E6

Encodings:

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See also:

Annotations for §27.8 and Ch.27

… ►A divisor $d$ of $k$ is called an induced modulus for $\chi$ if …

6: 27.10 Periodic Number-Theoretic Functions

§27.10 Periodic Number-Theoretic Functions

►If $k$ is a fixed positive integer, then a number-theoretic function $f$ is periodic (mod $k$) if … ►

27.10.3 $$g(m)=\frac{1}{k}\sum _{n=1}^{k}f(n){\mathrm{e}}^{-2\pi \mathrm{i}mn/k}.$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer, $f(n)$: function and $g(m)$: periodic function

Permalink:

http://dlmf.nist.gov/27.10.E3

Encodings:

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Annotations for §27.10 and Ch.27

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27.10.12 $$\chi \left(n\right)=\frac{G(1,\chi )}{k}\sum _{m=1}^k\overline{\chi }(m){\mathrm{e}}^{-2\pi \mathrm{i}mn/k}.$$

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Symbols:

$\chi \left(n\right)$: Dirichlet character, $G(n,\chi )$: Gauss sum, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $k$: positive integer, $m$: positive integer, $n$: positive integer and $\chi$: Dirichlet character

Permalink:

http://dlmf.nist.gov/27.10.E12

Encodings:

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See also:

Annotations for §27.10 and Ch.27

7: 27.21 Tables

§27.21 Tables

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8: 27.2 Functions

… ►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes. … ►Other examples of number-theoretic functions treated in this chapter are as follows. … ►

27.2.9 $$d\left(n\right)=\sum _{d|n}1$$

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Symbols:

$d_k\left(n\right)$: divisor function, $d$: positive integer and $n$: positive integer

A&S Ref:

24.3.3 I.A

Permalink:

http://dlmf.nist.gov/27.2.E9

Encodings:

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See also:

Annotations for §27.2(i), §27.2 and Ch.27

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Table 27.2.2: Functions related to division.

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$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$	$n$	$\varphi \left(n\right)$	$d\left(n\right)$	$\sigma \left(n\right)$
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5	4	2	6	18	6	6	39	31	30	2	32	44	20	6	84
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7	6	2	8	20	8	6	42	33	20	4	48	46	22	4	72
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9: 27.3 Multiplicative Properties

§27.3 Multiplicative Properties

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27.3.8 $$\varphi \left(m\right)\varphi \left(n\right)=\varphi \left(mn\right)\varphi \left(\left(m,n\right)\right)/\left(m,n\right).$$

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Symbols:

$\varphi \left(n\right)$: Euler’s totient, $\left(m,n\right)$: greatest common divisor (gcd), $m$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.3.E8

Encodings:

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See also:

Annotations for §27.3 and Ch.27

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10: 27.6 Divisor Sums

§27.6 Divisor Sums

►Sums of number-theoretic functions extended over divisors are of special interest. …