On Fractional Part Sums: A Mean-Square Asymptotics over Short Intervals

Abstract.

 This article is concerned with sums 𝒮(t) = ∑ _n  ψ(tf(n/t)) where ψ denotes, essentially, the fractional part minus ½, f is a C ⁴-function with f″ ≠ 0 throughout, summation being extended over an interval of order t. We establish an asymptotic formula for ∫ _T−Λ ^T+Λ(𝒮(t))²dt for any Λ = Λ(T) growing faster than log T.