Search for Periodic Variability in $γ$-ray Blazars Using \textit{Fermi}-LAT
Blazars are known to exhibit variability across a broad range of timescales. This behavior can include periodicity in their $γ$-ray emission, whose clear detection remains an ongoing challenge, partly due to the inherent stochasticity of the processes involved and also the lack of adequately well-sampled light curves. In this study, we perform a systematic search for periodicity in a selected sample of 24 $γ$-ray blazars using twelve years of Fermi-LAT data. The sample comprises the most promising candidates selected from a previous study, extending the light curves by three additional years, expanding the analyzed energy range from $>$1~GeV to $>$0.1~GeV to improve photon statistics, and enhancing the time-series analysis methodology. We incorporate upper-limit flux points in the analysis rather than discarding them, thereby preserving the temporal structure in the light curves. A suite of seven complementary time-series analysis methods is employed to ensure statistical robustness, including autoregressive models, representing a methodological advancement over the prior work. A further improvement is the explicit estimation of the look-elsewhere effect, which allows us to assess the global significance of any detected signals. The study is also supported by additional statistical treatments employed to minimize false detections and strengthen the reliability of the results. Our analysis reveals a hint of periodicity in PG 1553+113 with a global significance of $\approx$1.8$σ$. For the remaining sources in the sample, the re-evaluation of previously reported periodicities indicates that they are statistically consistent with arising from stochastic variability.