Measurements of long-range azimuthal anisotropies and associated Fourier coefficients for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mi>p</mml:mi></mml:mrow></mml:math> collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msqrt><mml:mi>s</mml:mi></mml:msqrt><mml:mo>=</mml:mo><mml:mn>5.02</mml:mn></mml:mrow></mml:math> and 13 TeV and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>p</mml:mi><mml:mo>+</mml:mo><mml:mtext>Pb</mml:mtext></mml:mrow></mml:math> collisions at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msqrt><mml:msub><mml:mi>s</mml:mi><mml:mi>NN</mml:mi></mml:msub></mml:msqrt><mml:mo>=</mml:mo><mml:mn>5.02</mml:mn></mml:mrow></mml:math> TeV with the ATLAS detector

ATLAS measurements of two-particle correlations are presented for $\sqrt{s}=5.02$ and 13 TeV $pp$ collisions and for $\sqrt{{s}_{\mathrm{NN}}}=5.02$ TeV $p+\text{Pb}$ collisions at the LHC. The correlation functions are measured as a function of relative azimuthal angle $\mathrm{\ensuremath{\Delta}}\ensuremath{\phi}$, and pseudorapidity separation $\mathrm{\ensuremath{\Delta}}\ensuremath{\eta}$, using charged particles detected within the pseudorapidity interval $|\ensuremath{\eta}|&lt;2.5$. Azimuthal modulation in the long-range component of the correlation function, with $|\mathrm{\ensuremath{\Delta}}\ensuremath{\eta}|&gt;2$, is studied using a template fitting procedure to remove a ``back-to-back'' contribution to the correlation function that primarily arises from hard-scattering processes. In addition to the elliptic, $\mathrm{cos}(2\mathrm{\ensuremath{\Delta}}\ensuremath{\phi})$, modulation observed in a previous measurement, the $pp$ correlation functions exhibit significant $\mathrm{cos}(3\mathrm{\ensuremath{\Delta}}\ensuremath{\phi})$ and $\mathrm{cos}(4\mathrm{\ensuremath{\Delta}}\ensuremath{\phi})$ modulation. The Fourier coefficients ${v}_{n,n}$ associated with the $\mathrm{cos}(n\mathrm{\ensuremath{\Delta}}\ensuremath{\phi})$ modulation of the correlation functions for $n=2--4$ are measured as a function of charged-particle multiplicity and charged-particle transverse momentum. The Fourier coefficients are observed to be compatible with $\mathrm{cos}(n\ensuremath{\phi})$ modulation of per-event single-particle azimuthal angle distributions. The single-particle Fourier coefficients ${v}_{n}$ are measured as a function of charged-particle multiplicity, and charged-particle transverse momentum for $n=2--4$. The integrated luminosities used in this analysis are, 64 ${\mathrm{nb}}^{\ensuremath{-}1}$ for the $\sqrt{s}=13$ TeV $pp$ data, 170 ${\mathrm{nb}}^{\ensuremath{-}1}$ for the $\sqrt{s}=5.02$ TeV $pp$ data, and 28 ${\mathrm{nb}}^{\ensuremath{-}1}$ for the $\sqrt{{s}_{\mathrm{NN}}}=5.02$ TeV $p+\text{Pb}$ data.