# multiplicative functions

(0.006 seconds)

## 1—10 of 85 matching pages

##### 1: 27.3 Multiplicative Properties
###### §27.3 Multiplicative Properties
Except for $\nu\left(n\right)$, $\Lambda\left(n\right)$, $p_{n}$, and $\pi\left(x\right)$, the functions in §27.2 are multiplicative, which means $f(1)=1$ and A function $f$ is completely multiplicative if $f(1)=1$ and …
##### 2: 27.20 Methods of Computation: Other Number-Theoretic Functions
To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). 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
Every multiplicative $f$ satisfies the identity …
27.4.2 $\sum_{n=1}^{\infty}f(n)=\prod_{p}(1-f(p))^{-1}.$
Euler products are used to find series that generate many functions of multiplicative number theory. 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)): …
##### 6: 17.8 Special Cases of ${{}_{r}\psi_{r}}$ Functions
17.8.2 ${{}_{1}\psi_{1}}\left({a\atop b};q,z\right)=\frac{\left(q,b/a,az,q/(az);q% \right)_{\infty}}{\left(b,q/a,z,b/(az);q\right)_{\infty}}.$
17.8.4 ${{}_{2}\psi_{2}}\left(b,c;aq/b,aq/c;q,-aq/(bc)\right)=\frac{\left(aq/(bc);q% \right)_{\infty}\left(aq^{2}/b^{2},aq^{2}/c^{2},q^{2},aq,q/a;q^{2}\right)_{% \infty}}{\left(aq/b,aq/c,q/b,q/c,-aq/(bc);q\right)_{\infty}},$
17.8.5 ${{}_{3}\psi_{3}}\left({b,c,d\atop q/b,q/c,q/d};q,\frac{q}{bcd}\right)=\frac{% \left(q,q/(bc),q/(bd),q/(cd);q\right)_{\infty}}{\left(q/b,q/c,q/d,q/(bcd);q% \right)_{\infty}},$
17.8.6 ${{}_{4}\psi_{4}}\left({-qa^{\frac{1}{2}},b,c,d\atop-a^{\frac{1}{2}},aq/b,aq/c,% aq/d};q,\frac{qa^{\frac{3}{2}}}{bcd}\right)=\frac{\left(aq,aq/(bc),aq/(bd),aq/% (cd),qa^{\frac{1}{2}}/b,qa^{\frac{1}{2}}/c,qa^{\frac{1}{2}}/d,q,q/a;q\right)_{% \infty}}{\left(aq/b,aq/c,aq/d,q/b,q/c,q/d,qa^{\frac{1}{2}},qa^{-\frac{1}{2}},% qa^{\frac{3}{2}}/(bcd);q\right)_{\infty}},$
17.8.7 ${{}_{6}\psi_{6}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},b,c,d,e\atop a^{% \frac{1}{2}},-a^{\frac{1}{2}},aq/b,aq/c,aq/d,aq/e};q,\frac{qa^{2}}{bcde}\right% )=\frac{\left(aq,aq/(bc),aq/(bd),aq/(be),aq/(cd),aq/(ce),aq/(de),q,q/a;q\right% )_{\infty}}{\left(aq/b,aq/c,aq/d,aq/e,q/b,q/c,q/d,q/e,qa^{2}/(bcde);q\right)_{% \infty}}.$
##### 7: 17.10 Transformations of ${{}_{r}\psi_{r}}$ Functions
17.10.1 ${{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,d/a,c/b,dq/(abz% );q\right)_{\infty}}{\left(z,d,q/b,cd/(abz);q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({a,abz/d\atop az,c};q,\frac{d}{a}\right),$
17.10.2 ${{}_{2}\psi_{2}}\left({a,b\atop c,d};q,z\right)=\frac{\left(az,bz,cq/(abz),dq/% (abz);q\right)_{\infty}}{\left(q/a,q/b,c,d;q\right)_{\infty}}{{}_{2}\psi_{2}}% \left({abz/c,abz/d\atop az,bz};q,\frac{cd}{abz}\right).$
17.10.3 ${{}_{8}\psi_{8}}\left({qa^{\frac{1}{2}},-qa^{\frac{1}{2}},c,d,e,f,aq^{-n},q^{-% n}\atop a^{\frac{1}{2}},-a^{\frac{1}{2}},aq/c,aq/d,aq/e,aq/f,q^{n+1},aq^{n+1}}% ;q,\frac{a^{2}q^{2n+2}}{cdef}\right)=\frac{\left(aq,q/a,aq/(cd),aq/(ef);q% \right)_{n}}{\left(q/c,q/d,aq/e,aq/f;q\right)_{n}}\*{{}_{4}\psi_{4}}\left({e,f% ,aq^{n+1}/(cd),q^{-n}\atop aq/c,aq/d,q^{n+1},ef/(aq^{n})};q,q\right),$
17.10.4 ${{}_{2}\psi_{2}}\left({e,f\atop aq/c,aq/d};q,\frac{aq}{ef}\right)=\frac{\left(% q/c,q/d,aq/e,aq/f;q\right)_{\infty}}{\left(aq,q/a,aq/(cd),aq/(ef);q\right)_{% \infty}}\*\sum_{n=-\infty}^{\infty}\frac{(1-aq^{2n})\left(c,d,e,f;q\right)_{n}% }{(1-a)\left(aq/c,aq/d,aq/e,aq/f;q\right)_{n}}\left(\frac{qa^{3}}{cdef}\right)% ^{n}q^{n^{2}}.$
17.10.5 $\frac{\left(aq/b,aq/c,aq/d,aq/e,q/(ab),q/(ac),q/(ad),q/(ae);q\right)_{\infty}}% {\left(fa,ga,f/a,g/a,qa^{2},q/a^{2};q\right)_{\infty}}\*{{}_{8}\psi_{8}}\left(% {qa,-qa,ba,ca,da,ea,fa,ga\atop a,-a,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g};q,\frac{q^{% 2}}{bcdefg}\right)=\frac{\left(q,q/(bf),q/(cf),q/(df),q/(ef),qf/b,qf/c,qf/d,qf% /e;q\right)_{\infty}}{\left(fa,q/(fa),aq/f,f/a,g/f,fg,qf^{2};q\right)_{\infty}% }\*{{}_{8}\phi_{7}}\left({f^{2},qf,-qf,fb,fc,fd,fe,fg\atop f,-f,fq/b,fq/c,fq/d% ,fq/e,fq/g};q,\frac{q^{2}}{bcdefg}\right)+\operatorname{idem}\left(f;g\right).$
##### 8: 17.7 Special Cases of Higher ${{}_{r}\phi_{s}}$ Functions
17.7.2 ${{}_{2}\phi_{2}}\left({a^{2},b^{2}\atop abq^{\frac{1}{2}},-abq^{\frac{1}{2}}};% q,-q\right)=\frac{\left(a^{2}q,b^{2}q;q^{2}\right)_{\infty}}{\left(q,a^{2}b^{2% }q;q^{2}\right)_{\infty}}.$
17.7.4 ${{}_{3}\phi_{2}}\left({a,b,q^{-n}\atop c,abq^{1-n}/c};q,q\right)=\frac{\left(c% /a,c/b;q\right)_{n}}{\left(c,c/(ab);q\right)_{n}}.$
17.7.12 ${{}_{4}\phi_{3}}\left({a,aq,b^{2}q^{2n},q^{-2n}\atop b,bq,a^{2}q^{2}};q^{2},q^% {2}\right)=\frac{a^{n}\left(-q,b/a;q\right)_{n}}{\left(-aq,b;q\right)_{n}}.$
17.7.13 ${{}_{4}\phi_{3}}\left({a,aq,b^{2}q^{2n-2},q^{-2n}\atop b,bq,a^{2}};q^{2},q^{2}% \right)=\frac{a^{n}\left(-q,b/a;q\right)_{n}(1-bq^{n-1})}{\left(-a,b;q\right)_% {n}(1-bq^{2n-1})}.$
##### 10: 17.6 ${{}_{2}\phi_{1}}$ Function
17.6.5 ${{}_{2}\phi_{1}}\left({a,b\atop aq/b};q,-q/b\right)=\frac{\left(-q;q\right)_{% \infty}\left(aq,\ifrac{aq^{2}}{b^{2}};q^{2}\right)_{\infty}}{\left(-q/b,aq/b;q% \right)_{\infty}},$ $|b|>|q|$.
17.6.6 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(b,az;q\right)_{% \infty}}{\left(c,z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({c/b,z\atop az};q,b% \right),$ $|z|<1,|b|<1$.
17.6.7 ${{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)=\frac{\left(c/b,bz;q\right)_{% \infty}}{\left(c,z;q\right)_{\infty}}{{}_{2}\phi_{1}}\left({\ifrac{abz}{c},b% \atop bz};q,c/b\right),$ $|z|<1,|c|<|b|$.
17.6.10 $(1-z){{}_{2}\phi_{1}}\left({q,aq\atop bq};q,z\right)=\sum_{n=0}^{\infty}\frac{% \left(b/a;q\right)_{n}(-az)^{n}q^{(n^{2}+n)/2}}{\left(bq,zq;q\right)_{n}},$ $|z|<1$.