Higher-order NLO initial state QED radiative corrections to $e^+e^-$ annihilation revisited

Radiative corrections due to initial state radiation in electron-positron annihilation are calculated within the QED structure function approach. Results are shown in the next-to-leading logarithmic approximation up to $O(\alpha^4 L^3)$ order, where $L=\ln(s/m_e^2)$ is the large logarithm. Several mistakes in previous calculations are corrected. The results are relevant for future high-precision experiments at $e^+e^-$ colliders.