デデキントのイータ関数
イータ関数はテータ関数で表される。オイラーの分割恒等式を用いて
η 3 ( τ ) = e π i τ / 4 ∏ m = 1 ∞ ( 1 − e 2 π i τ m ) 3 = e π i τ / 4 ∏ m = 1 ∞ ( 1 − e 2 π i τ m ) 3 ( 1 + e 2 π i τ m ) 2 ( 1 − e 2 π i τ ( 2 m − 1 ) ) 2 = e π i τ / 4 ∏ m = 1 ∞ ( 1 − e 2 π i τ m ) 3 ( 1 + e 2 π i τ m ) 2 ( 1 + e ( 2 m − 1 ) π i τ ) 2 ( 1 − e ( 2 m − 1 ) π i τ ) 2 = 1 2 θ 2 ( 0 , τ ) θ 3 ( 0 , τ ) θ 4 ( 0 , τ ) {\displaystyle {\begin{aligned}\eta ^{3}\left(\tau \right)&=e^{\pi {i}\tau /4}\prod _{m=1}^{\infty }\left(1-e^{2\pi {i}\tau {m}}\right)^{3}\\&=e^{\pi {i}\tau /4}\prod _{m=1}^{\infty }\left(1-e^{2\pi {i}\tau {m}}\right)^{3}\left(1+e^{2\pi {i}\tau {m}}\right)^{2}\left(1-e^{2\pi {i}\tau {(2m-1)}}\right)^{2}\\&=e^{\pi {i}\tau /4}\prod _{m=1}^{\infty }\left(1-e^{2\pi {i}\tau {m}}\right)^{3}\left(1+e^{2\pi {i}\tau {m}}\right)^{2}\left(1+e^{(2m-1)\pi {i}\tau }\right)^{2}\left(1-e^{(2m-1)\pi {i}\tau }\right)^{2}\\&={\frac {1}{2}}\vartheta _{2}\left(0,\tau \right)\vartheta _{3}\left(0,\tau \right)\vartheta _{4}\left(0,\tau \right)\\\end{aligned}}}