チェビシェフ関数
006788 x ln  x {\displaystyle |\vartheta (x)-x|\leq 0.006788{\frac {x}{\ln x}}} for x ≥ 10544111 , {\displaystyle x\geq 10544111,}
| ψ ( x ) − x | ≤ 0.006409 x ln  x {\displaystyle |\psi (x)-x|\leq 0.006409{\frac {x}{\ln x}}} for x ≥ e 2 2 , {\displaystyle x\geq e^{2}2,}
ψ ( x ) < ( 1 + 1 36260 ) x {\displaystyle \psi (x)<\left(1+{\frac {1}{36260}}\right)x} for x > 0 , {\displaystyle x>0,}
ψ ( x ) − θ ( x ) > 0.9999 x {\displaystyle \psi (x)-\vartheta (x)>0.9999{\sqrt {x}}} for x ≥ 121 , {\displaystyle x\geq 121,}
0.9999 x < ψ ( x ) − θ ( x ) < 1.