チェビシェフ関数
第二チェビシェフ関数を補正した関数
ψ 0 ( x ) = lim h → 0 1 2 ( ψ ( x + h ) + ψ ( x − h ) ) = 1 2 ( ∑ n ≤ x Λ ( n ) + ∑ n < x Λ ( n ) ) = { ψ ( x ) − 1 2 Λ ( x ) x = 2 , 3 , 4 , 5 , 7 , 8 , 9 , 11 , 13 , 16 , ... ψ ( x ) otherwise. {\displaystyle \psi _{0}(x)=\lim _{h\rightarrow 0}{\frac {1}{2}}\left(\psi (x+h)+\psi (x-h)\right)={\frac {1}{2}}\left(\sum _{n\leq x}\Lambda (n)+\sum _{n<x}\Lambda (n)\right)={\begin{cases}\psi (x)-{\frac {1}{2}}\Lambda (x)&x=2,3,4,5,7,8,9,11,13,16,\dots \\\psi (x)&{\mbox{otherwise.}}\end{cases}}}