オイラーの定数
オイラーの定数の値は以下の級数で得られる.
γ = ∑ n = 2 ∞ ( − 1 ) n ζ ( n ) n {\displaystyle \gamma =\sum _{n=2}^{\infty }{\frac {(-1)^{n}\zeta (n)}{n}}}
γ = 1 − ∑ n = 2 ∞ ζ ( n ) − 1 n {\displaystyle \gamma =1-\sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}}
γ = ∑ n = 2 ∞ ( n − 1 ) ( ζ ( n ) − 1 ) n {\displaystyle \gamma =\sum _{n=2}^{\infty }{\frac {(n-1)(\zeta (n)-1)}{n}}}
γ = ln  2 − ∑ n = 1 ∞ ζ ( 2 n + 1 ) 2 2 n ( 2 n + 1 ) {\displaystyle \gamma =\ln 2-\sum _{n=1}^{\infty }{\frac {\zeta (2n+1)}{2^{2n}(2n+1)}}}