Abstract

We establish a new lower bound for the average local discrepancy of the Farey sequence. First, we prove a general lower-bound formula for arbitrary sequences in the unit interval in terms of their gap distribution, namely the gaps larger than the average gap and the minimum gap. We then apply this result to the Farey sequence and obtain an explicit lower bound of logarithmic order. This complements the existing literature, which mainly focuses on upper bounds, and provides a new constraint on a quantity that is closely related to the Franel–Landau criterion for the Riemann Hypothesis. Finally, we examine the effect of gap permutations on the average local discrepancy and propose a conjecture for the corresponding mean value over all such permutations.

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