The study of the gluon distribution function and reduced cross section behavior using the proton structure function

The behavior of the gluon distribution function and the reduced cross section considered from the proton structure function and its derivatives at low values of $x$. These behaviors studied and compared with the experimental data. These results are augmented by including an additional higher-twist term in the description of the nonlinear correction. This additional term, modified nonlinear correction, improves the description of the reduced cross sections significantly at low values of $Q^{2}$. We discuss, furthermore, how this behavior can be determine the reduced cross section with respect to the proton parameterization at high-y values. The resulting predictions for $\sigma_{r}$ suggest that further corrections are required for $Q^{2}$ less than about $3~\mathrm{GeV}^{2}$.