ラマヌジャンのタウ函数
τ ( n ) ≡ n σ 9 ( n ) mod 7 2 for n ≡ 3 , 5 , 6 mod 7 {\displaystyle \tau (n)\equiv n\sigma _{9}(n)\ {\bmod {\ }}7^{2}{\text{ for }}n\equiv 3,5,6\ {\bmod {\ }}7}
τ ( n ) ≡ σ 11 ( n ) mod 691. {\displaystyle \tau (n)\equiv \sigma _{11}(n)\ {\bmod {\ }}691.}
素数 p ≠ 23 に対して,次が成り立つ:
τ ( p ) ≡ 0 mod 23 if ( p 23 ) = − 1 {\displaystyle \tau (p)\equiv 0\ {\bmod {\ }}23{\text{ if }}\left({\frac {p}{23}}\right)=-1}
τ ( p ) ≡ σ 11 ( p ) mod 23 2 if p is of the form a 2 + 23 b 2 {\displaystyle \tau (p)\equiv \sigma _{11}(p)\ {\bmod {\ }}23^{2}{\text{ if }}p{\text{ is of the form }}a^{2}+23b^{2}}
τ ( p ) ≡ − 1 mod 23 otherwise . {\displaystyle \tau (p)\equiv -1\ {\bmod {\ }}23{\text{ otherwise}}.}