number-theoretic functions

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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

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3: 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. … ►

27.5.3 $$g(n)=\sum _{d|n}f(d)⟺f(n)=\sum _{d|n}g(d)\mu \left(\frac{n}{d}\right).$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $d$: positive integer, $n$: positive integer, $f$: function and $g$: function

A&S Ref:

24.3.1 II.C

Permalink:

http://dlmf.nist.gov/27.5.E3

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See also:

Annotations for §27.5 and Ch.27

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4: 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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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: 27.6 Divisor Sums

§27.6 Divisor Sums

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

9: 27.7 Lambert Series as Generating Functions

§27.7 Lambert Series as Generating Functions

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10: 27.9 Quadratic Characters

§27.9 Quadratic Characters

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