Differential production cross sections of prompt J/ψ and ψ(2S) charmonium and Υ(nS) (n = 1, 2, 3) bottomonium states are measured in proton-proton collisions at √ s = 13 TeV, with data collected by the CMS detector at the LHC, corresponding to an integrated luminosity of 2.3 fb −1 for the J/ψ and 2.7 fb −1 for the other mesons. The five quarkonium states are reconstructed in the dimuon decay channel, for dimuon rapidity |y| < 1.2. The double-differential cross sections for each state are measured as a function of y and transverse momentum, and compared to theoretical expectations. In addition, ratios are presented of cross sections for prompt ψ(2S) to J/ψ, Υ(2S) to Υ(1S), and Υ(3S) to Υ(1S) production.The analysis uses dimuon events collected in pp collisions at √ s = 13 TeV with the CMS detector. The central feature of the CMS apparatus is a superconducting solenoid of 6 m internal diameter, providing a magnetic field of 3.8 T. Within the solenoid volume are a silicon pixel and strip tracker, a lead tungstate crystal electromagnetic calorimeter, and a brass and scintillator hadron calorimeter, each composed of a barrel and two endcap sections. Forward calorimeters extend the pseudorapidity (η) coverage provided by the barrel and endcap detectors. Muons are detected in gas-ionization chambers embedded in the steel flux-return yoke outside the solenoid [37]. A more detailed description of the CMS detector, together with a definition of the coordinate system used and the relevant kinematic variables, can be found in Ref. [38].The data were collected using a multilevel trigger system [39]. The first level (L1), made of custom hardware processors providing coarse momentum information, requires two muons within the range |η| < 1.6 without requesting an explicit p T threshold on the individual muons. Second (L2) and third (L3) levels, collectively known as the HLT (High-Level Trigger), are implemented in software. At these levels, the muon selection is refined, then opposite-charge muon candidates are paired and required to have an invariant mass in the regions 2.9-3.3, 3.35-4.05, or 8.5-11 GeV for the J/ψ, ψ(2S), and Υ(nS), respectively. The dimuon p T is required to be above 9.9 GeV for the J/ψ and above 7.9 GeV for the remaining states. For all five states, the dimuon rapidity is restricted to |y| < 1.25. A fit of the positions and momenta of the two muon candidates to a common vertex is performed, and the fit χ 2 probability is required to be above 0.5%. The sample collected with these triggers has a total integrated luminosity of 2.3 fb −1 for the J/ψ and 2.7 fb −1 for the other mesons. The lower value for the J/ψ is the consequence of the trigger prescaling that was applied to limit the rate during part of the data taking, when the instantaneous luminosity increased.When reconstructing the five states offline, further requirements are applied: only muons with p µ T > 4.5 GeV in the range |η µ | < 0.3, or p µ T > 4.0 GeV in the range 0.3 < |η µ | < 1.4 are selected. The muons have to match the triggered pair and b...