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) = Sigma(kappa/lambda is an element of) f(kappa/lambda), F-n,F-sigma(f) = Sigma(kappa/lambda is an element of Fn) 1/kappa sigma lambda sigma f(kappa/lambda), 0 < sigma <= 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.

Automated summary

This paper presents sharp effective estimates for Farey sums and proves an unconditional lower bound for Farey sums.