The STAR Collaboration reports measurements of the longitudinal double-spin asymmetry, $A_{LL}$, for neutral pions produced at forward directions in polarized proton-proton collisions, at a center-of-mass energy of $510$ GeV. Results are given for transverse momenta in the range $2<p_{T}<10$ GeV/$c$ within two regions of pseudorapidity that span $2.65<eta<3.9$. These results are sensitive to the polarized gluon parton distribution function, $Delta g(x)$, down to the region of Bjorken $x sim 10^{-3}$. The asymmetries observed are less than $pm 5 cdot 10^{-3}$ in magnitude, and will help constrain the contribution to the spin of the proton from polarized gluons at low $x$, when combined with other measurements as part of a global analysis.
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