This paper focuses on time series with long-memory and suggests using an alternative periodogram, called M-periodogram, which is obtained by relating the periodogram to a regression problem and then using an M-estimator for the coefficients of the regression model. The asymptotic properties of this novel M-periodogram are established and its empirical properties are investigated for finite samples under different scenarios. Furthermore, in addition to being an appealing alternative periodogram for long-memory time series, it is also resistant to additive outliers. We investigate the robustness performance of the estimator through simulation. As a practical application, the paper investigates the effect of atypical observations in air pollution data, namely daily Particulate Matter (PM10PM10) observations. Besides the importance of modelling and forecasting this pollutant, the PM10PM10 series presents, in general, interesting features such as seasonal poles, asymmetry, and also high levels of pollution which can be regarded as atypical observations in the context of this work.