# completely multiplicative functions

(0.003 seconds)

## 1—10 of 17 matching pages

##### 1: 27.3 Multiplicative Properties
A function $f$ is completely multiplicative if $f(1)=1$ and …
##### 2: 27.20 Methods of Computation: Other Number-Theoretic Functions
For a completely multiplicative function we use the values at the primes together with (27.3.10). …
##### 3: 27.4 Euler Products and Dirichlet Series
The completely multiplicative function $f(n)=n^{-s}$ gives the Euler product representation of the Riemann zeta function $\zeta\left(s\right)$25.2(i)): …
##### 4: 27.8 Dirichlet Characters
If $k$ $(>1)$ is a given integer, then a function $\chi\left(n\right)$ is called a Dirichlet character (mod $k$) if it is completely multiplicative, periodic with period $k$, and vanishes when $\left(n,k\right)>1$. …
##### 5: 22.5 Special Values
Table 22.5.1 gives the value of each of the 12 Jacobian elliptic functions, together with its $z$-derivative (or at a pole, the residue), for values of $z$ that are integer multiples of $K$, $iK^{\prime}$. …
##### 6: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
Such orthonormal sets are called complete. … , of unit multiplicity, unless otherwise stated. …and completeness implies … and completeness relation …
###### §22.8(iii) Special Relations Between Arguments
If sums/differences of the $z_{j}$’s are rational multiples of $K\left(k\right)$, then further relations follow. …
##### 8: 23.1 Special Notation
(For other notation see Notation for the Special Functions.)
 $\mathbb{L}$ lattice in $\mathbb{C}$. … complete elliptic integrals (§19.2(i)). … set of all integer multiples of $n$. …
The main functions treated in this chapter are the Weierstrass $\wp$-function $\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)$; the elliptic modular function $\lambda\left(\tau\right)$; Klein’s complete invariant $J\left(\tau\right)$; Dedekind’s eta function $\eta\left(\tau\right)$. …
##### 9: 27.2 Functions
###### §27.2(i) Definitions
They tend to thin out among the large integers, but this thinning out is not completely regular. … This is Jordan’s function. … This is Liouville’s function. …
##### 10: Bibliography K
• M. K. Kerimov and S. L. Skorokhodov (1985c) Calculation of the multiple zeros of the derivatives of the cylindrical Bessel functions $J_{\nu}(z)$ and $Y_{\nu}(z)$ . Zh. Vychisl. Mat. i Mat. Fiz. 25 (12), pp. 1749–1760, 1918.
• M. K. Kerimov and S. L. Skorokhodov (1986) On multiple zeros of derivatives of Bessel’s cylindrical functions. Dokl. Akad. Nauk SSSR 288 (2), pp. 285–288 (Russian).
• M. K. Kerimov and S. L. Skorokhodov (1987) On the calculation of the multiple complex roots of the derivatives of cylindrical Bessel functions. Zh. Vychisl. Mat. i Mat. Fiz. 27 (11), pp. 1628–1639, 1758.
• M. K. Kerimov and S. L. Skorokhodov (1988) Multiple complex zeros of derivatives of the cylindrical Bessel functions. Dokl. Akad. Nauk SSSR 299 (3), pp. 614–618 (Russian).
• S. Koumandos and M. Lamprecht (2010) Some completely monotonic functions of positive order. Math. Comp. 79 (271), pp. 1697–1707.