라마누잔 타우 함수

라마누잔 타우 함수(Ramanujan Tau Function) ${\displaystyle \;\;\tau (n)}$는 타우 함수(Tau Function)로도 불린다.^[1]

${\displaystyle n}$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
${\displaystyle \tau (n)}$	1	−24	252	−1472	4830	−6048	−16744	84480	−113643	−115920	534612	−370944	−577738	401856	1217160	987136

${\displaystyle \qquad =1x^{1}-24x^{2}+252x^{3}-1472x^{4}+4830x^{5}-6048x^{6}+....}$

${\displaystyle \qquad =x(1-3x^{1}+5x^{3}-7x^{6}+....)^{8}}$

${\displaystyle \qquad =x\prod _{n=1}^{\infty }(1-x^{n})^{24}}$

${\displaystyle \qquad =\sum _{n=1}^{\infty }\tau (n)x^{n}}$

${\displaystyle \tau (n)(n-1)=\sum _{k=1}^{b_{n}}(-1)^{k+1}(2k+1)\left(n-1-9K\right)\tau \left(n-K\right)}$

${\displaystyle b_{n}={{{\sqrt {8n+1}}-1} \over 2}}$

${\displaystyle K={{k(k+1)} \over 2}}$

따라서,

${\displaystyle \tau (n)(n-1)=\sum _{k=1}^{b_{n}}(-1)^{k+1}(2k+1)\left(n-1-9K\right)\tau \left(n-K\right)}$

${\displaystyle \sum _{n=1}^{\infty }\tau (n)x^{n}=\eta (\tau )^{24}=\Delta (\tau )}$

${\displaystyle x=e^{2\pi i\tau }}$

${\displaystyle \eta (\tau )}$ 데데킨트 에타 함수

• 약수함수
• L-함수