# number-theoretic significance

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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. …
##### 4: 27.5 Inversion Formulas
###### §27.5 Inversion Formulas
The set of all number-theoretic functions $f$ with $f(1)\neq 0$ forms an abelian group under Dirichlet multiplication, with the function $\left\lfloor 1/n\right\rfloor$ in (27.2.5) as identity element; see Apostol (1976, p. 129). …Generating functions yield many relations connecting number-theoretic functions. …
27.5.2 $\sum_{d\mathbin{|}n}\mu\left(d\right)=\left\lfloor\frac{1}{n}\right\rfloor,$
##### 5: 27.8 Dirichlet Characters
###### §27.8 Dirichlet Characters
If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation
27.8.6 $\sum_{r=1}^{\phi\left(k\right)}\chi_{r}\left(m\right)\overline{\chi}_{r}(n)=% \begin{cases}\phi\left(k\right),&m\equiv n\pmod{k},\\ 0,&\mbox{otherwise}.\end{cases}$
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 $g(m)=\dfrac{1}{k}\sum_{n=1}^{k}f(n)e^{-2\pi\mathrm{i}mn/k}.$
27.10.12 $\chi\left(n\right)=\frac{G\left(1,\chi\right)}{k}\sum_{m=1}^{k}\overline{\chi}% (m)e^{-2\pi\mathrm{i}mn/k}.$
##### 9: 27.6 Divisor Sums
###### §27.6 Divisor Sums
Sums of number-theoretic functions extended over divisors are of special interest. …