Let \(\lambda_ j\) be the eigenvalues of the positive elliptic pseudodifferential operator of order \(m>0\) on a compact closed d- dimensional \(C^{\infty}\)-manifold, \(N(\lambda)=\#\{j:\lambda_ j\leq \lambda^ m\}\). It is shown that for each \(\epsilon >0\) \[ C_ 0(\lambda +\epsilon)^ d+C_ 1\lambda^{d-1}+Q(\lambda +\epsilon)\lambda^{d-1}+O(\lambda^{d-1})\geq N(\lambda)\geq C_ 0(\lambda -\epsilon)^ d+c_ 1\lambda^{d-1}+Q(\lambda - \epsilon)\lambda^{d-1}+O(\lambda^{d-1}), \] where \(C_ 0\) and \(C_ 1\) are standard Weyl constants, Q(\(\mu)\) is some bounded function on \({\mathbb{R}}^ 1\). The function Q(\(\mu)\) describes the influence of periodic bicharacteristics on the asymptotics of N(\(\lambda)\). Under assumption of simple reflection of bicharacteristics this result is valid for differential operators on compact manifolds with boundary too.