How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces

In this paper, we establish a strategy for the calculation of the proportion of everywhere locally soluble diagonal hypersurfaces of [Formula: see text] of fixed degree. Our strategy is based on the product formula established by Bright, Browning and Loughran. Their formula reduces the problem into the calculation of the proportions of [Formula: see text]-soluble diagonal hypersurfaces for all places [Formula: see text]. As worked examples, we carry out our strategy in the cases of quadratic and cubic hypersurfaces. As a consequence, we prove that around [Formula: see text] of diagonal cubic fourfolds have [Formula: see text]-rational points under a hypothesis on the Brauer–Manin obstruction.