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The measurements of the production of prompt D0, D+, D∗+, and D+s mesons in proton–proton (pp) collisions at s√=5.02 TeV with the ALICE detector at the Large Hadron Collider (LHC) are reported. D mesons were reconstructed at mid-rapidity (|y|<0.5) via their hadronic decay channels D0→K−π+, D+→K−π+π+, D∗+→D0π+→K−π+π+, D+s→ϕπ+→K+K−π+, and their charge conjugates. The production cross sections were measured in the transverse momentum interval 0<pT<36 GeV/c for D0, 1<pT<36 GeV/c for D+ and D∗+, and in 2<pT<24 GeV/c for D+s mesons. Thanks to the higher integrated luminosity, an analysis in finer pT bins with respect to the previous measurements at s√=7 TeV was performed, allowing for a more detailed description of the cross-section pT shape. The measured pT-differential production cross sections are compared to the results at s√=7 TeV and to four different perturbative QCD calculations. Its rapidity dependence is also tested combining the ALICE and LHCb measurements in pp collisions at s√=5.02 TeV. This measurement will allow for a more accurate determination of the nuclear modification factor in p–Pb and Pb–Pb collisions performed at the same nucleon–nucleon centre-of-mass energy.