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Abstract Pulsar timing array (PTA) searches for gravitational waves (GWs) aim to detect a characteristic correlation pattern in the timing residuals of galactic millisecond pulsars. This pattern is described by the PTA overlap reduction function (ORF) ${\mathrm{\Gamma }}_{ab}\left({\xi }_{ab}\right)$, which is known as the Hellings–Downs (HD) curve in general relativity (GR). In theories of modified gravity, the HD curve often receives corrections. Assuming, e.g. a subluminal GW phase velocity, one finds a drastically enhanced ORF in the limit of small angular separations between pulsaraand pulsarbin the sky, ${\xi }_{ab}\to 0$. In particular, working in harmonic space and performing an approximate resummation of all multipole contributions, the auto correlation coefficientΓ_aaseems to diverge. In this paper, we confirm that this divergence is unphysical and provide an exact and analytical expression forΓ_aain dependence of the pulsar distanceL_aand the GW phase velocity $v_{\mathrm{ph}}$. In the GR limit and assuming a large pulsar distance, our expression reduces to ${\mathrm{\Gamma }}_{aa}=1$. In the case of subluminal phase velocity, we show that the regularization of the naive divergent result is a finite-distance effect, meaning thatΓ_aascales linearly withfL_a, wherefis the GW frequency. For superluminal phase velocity (subluminal group velocity), which is relevant in the case of massive gravity, we correct an earlier analytical result forΓ_ab. Our results pave the way for fitting modified-gravity theories with nonstandard phase velocity to PTA data, which requires a proper understanding of the auto correlation coefficientΓ_aa.