We improve the current upper and lower bounds for the normal order of the Erdős–Hooley Δ–function obtaining, for almost all integers n , the inequalities where the exponent γ := (log 2)/log((1−1/log 27)/(1 − 1/log 3)) ≈ 0.33827 is conjectured to be optimal.
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Maier et al. (2009) studied this question.