On Farey sequence and quadratic Farey sums

Let F n be the Farey series of order n ≥ 1 . We obtain sharp effective estimates of the Farey sums F n ( f ) = ∑ κ λ ∈ F n f ( κ λ ) , F n , σ ( f ) = ∑ κ λ ∈ F n 1 κ σ λ σ f ( κ λ ) , 0 < σ ≤ 1 , for 1-periodic functions f satisfying weak local regularity assumption of Dini’s type at rational points of ]0, 1[. We also prove an unconditional lower bound for Farey sums.