# completely multiplicative

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##### 1: 27.3 Multiplicative Properties
A function $f$ is completely multiplicative if $f(1)=1$ and … If $f$ is completely multiplicative, then (27.3.2) becomes …
##### 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
If $f(n)$ is completely multiplicative, then each factor in the product is a geometric series and the Euler product becomes … 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 $\left(n,k\right)>1$. …
If sums/differences of the $z_{j}$’s are rational multiples of $K\left(k\right)$, then further relations follow. …
##### 6: 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 $K$, $iK^{\prime}$. …
##### 7: 3.2 Linear Algebra
To an eigenvalue of multiplicity $m$, there correspond $m$ linearly independent eigenvectors provided that $\mathbf{A}$ is nondefective, that is, $\mathbf{A}$ has a complete set of $n$ linearly independent eigenvectors. …
##### 8: 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 …
##### 9: 23.1 Special Notation
 $\mathbb{L}$ 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 $\eta\left(\tau\right)$. …
##### 10: Errata
• Equation (17.6.1)
17.6.1 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,\ifrac{c}{(ab)}\right)=\frac{\left(c/a,c/% b;q\right)_{\infty}}{\left(c,c/(ab);q\right)_{\infty}},$ $|c|<|ab|$

The constraint $|c|<|ab|$ was added.

• Equation (17.11.2)
17.11.2 $\Phi^{(2)}\left(a;b,b^{\prime};c,c^{\prime};q;x,y\right)=\frac{\left(b,ax;q% \right)_{\infty}}{\left(c,x;q\right)_{\infty}}\sum_{n,r\geqq 0}\frac{\left(a,b% ^{\prime};q\right)_{n}\left(c/b,x;q\right)_{r}b^{r}y^{n}}{\left(q,c^{\prime};q% \right)_{n}\left(q;q\right)_{r}\left(ax;q\right)_{n+r}}$

The factor ${\left(q\right)_{r}}$ originally used in the denominator has been corrected to be $\left(q;q\right)_{r}$.

• Equation (17.4.6)

The multi-product notation $\left(q,c;q\right)_{m}\left(q,c^{\prime};q\right)_{n}$ in the denominator of the right-hand side was used.

• Equations (17.2.22) and (17.2.23)
17.2.22 $\frac{\left(qa^{\frac{1}{2}},-qa^{\frac{1}{2}};q\right)_{n}}{\left(a^{\frac{1}% {2}},-a^{\frac{1}{2}};q\right)_{n}}=\frac{\left(aq^{2};q^{2}\right)_{n}}{\left% (a;q^{2}\right)_{n}}=\frac{1-aq^{2n}}{1-a}$
17.2.23 $\frac{\left(qa^{\frac{1}{k}},q\omega_{k}a^{\frac{1}{k}},\dots,q\omega_{k}^{k-1% }a^{\frac{1}{k}};q\right)_{n}}{\left(a^{\frac{1}{k}},\omega_{k}a^{\frac{1}{k}}% ,\dots,\omega_{k}^{k-1}a^{\frac{1}{k}};q\right)_{n}}=\frac{\left(aq^{k};q^{k}% \right)_{n}}{\left(a;q^{k}\right)_{n}}=\frac{1-aq^{kn}}{1-a}$

The numerators of the leftmost fractions were corrected to read $\left(qa^{\frac{1}{2}},-qa^{\frac{1}{2}};q\right)_{n}$ and $\left(qa^{\frac{1}{k}},q\omega_{k}a^{\frac{1}{k}},\dots,q\omega_{k}^{k-1}a^{% \frac{1}{k}};q\right)_{n}$ instead of $\left(qa^{\frac{1}{2}},-aq^{\frac{1}{2}};q\right)_{n}$ and $\left(aq^{\frac{1}{k}},q\omega_{k}a^{\frac{1}{k}},\dots,q\omega_{k}^{k-1}a^{% \frac{1}{k}};q\right)_{n}$, respectively.

Reported 2017-06-26 by Jason Zhao.

• References

An addition was made to the Software Index to reflect a multiple precision (MP) package written in C++ which uses a variety of different MP interfaces. See Kormanyos (2011).