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GaN thin layers were grown on stripe-patterned silicon (111) substrates with an AlN buffer layer by using metal organic chemical vapor deposition. The Si substrates were structured with various pattern sizes of 2.95, 7.85, and 18.2 $\mu$m by using a conventional photolithography and inductively coupled plasma etching process. The GaN layer on the stripe-patterned Si extends vertically and laterally during the growth. Photoluminescence (PL) spectra show a band-edge emission at 3.37 eV corresponding to the exciton recombination and a blue one around 2.8 eV corresponding to the transition associated with intrinsic defects such as oxygen and hydrogenated gallium vacancies. From the $\mu$-PL spectra, the intensity of the band-edge emission from the laterally overgrown region of GaN layer is stronger than that from the vertical one, and the blue emission is not observed for the lateral one. Especially, when the pattern size was 7.85 $\mu$m, a 7.02-fold increase in PL intensity for the laterally overgrown region was obtained in comparison with that for the vertical one. We conclude that the enhancement of the optical properties for lateral overgrowth of GaN layers depends on the pattern size of the Si substrate and results from a decrease in the dislocation density.