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Measurements of two-particle correlation functions and the first fiveazimuthal harmonics, $v_1$ to $v_5$, are presented, using 28 $\mathrm{nb}^{-1}$of $p$+Pb collisions at a nucleon-nucleon center-of-mass energy of$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV measured with the ATLAS detector at the LHC.Significant long-range ``ridge-like'' correlations are observed for pairs withsmall relative azimuthal angle ($|\Delta\phi|<\pi/3$) and back-to-back pairs($|\Delta\phi|>2\pi/3$) over the transverse momentum range $0.4<p_{\rm T}<12$GeV and in different intervals of event activity. The event activity is definedby either the number of reconstructed tracks or the total transverse energy onthe Pb-fragmentation side. The azimuthal structure of such long-rangecorrelations is Fourier decomposed to obtain the harmonics $v_n$ as a functionof $p_{\rm T}$ and event activity. The extracted $v_n$ values for $n=2$ to $5$decrease with $n$. The $v_2$ and $v_3$ values are found to be positive in themeasured $p_{\rm T}$ range. The $v_1$ is also measured as a function of $p_{\rmT}$ and is observed to change sign around $p_{\rm T}\approx 1.5-2.0$ GeV andthen increase to about 0.1 for $p_{\rm T}>4$ GeV. The $v_2(p_{\rm T})$,$v_3(p_{\rm T})$ and $v_4(p_{\rm T})$ are compared to the $v_n$ coefficients inPb+Pb collisions at $\sqrt{s_{\mathrm{NN}}} =2.76$ TeV with similar eventmultiplicities. Reasonable agreement is observed after accounting for thedifference in the average $p_{\rm T}$ of particles produced in the twocollision systems.

Measurement of long-range pseudorapidity correlations and azimuthal  harmonics in $\sqrt{s_{\mathrm{NN}}}$ = 5.02 TeV proton-lead collisions with  the ATLAS detector