AbstractIn this article we consider sums S(t) = Σnψ (tf(n/t)), where ψ denotes, essentially, the fractional part minus ½ f is a C4-function with f″ non-vanishing, and summation is extended over an interval of order t. For the mean-square of S(t), an asymptotic formula is established. If f is algebraic this can be sharpened by the indication of an error term.
We give a proof for the approximate functional equation for exponential sums related to holomorphic cusp forms and derive an upper bound for the error term.
We study Δ(x; ϕ), the error term in the asymptotic formula for
[sum ]n[les ]xcn, where
the cns are generated by the
Rankin–Selberg series. Our main tools are Voronoï-type
formulae. First we reduce the evaluation of Δ(x; ϕ)
to that of Δ1(x; ϕ), the error term of the weighted sum
[sum ]n[les ]x(x−n)cn.
Then we prove an upper bound and a sharp mean square formula for
Δ1(x; ϕ), by applying the Voronoï formula of
Meurman's type. We also prove that an improvement of the error term in the mean square
formula would imply an improvement of the upper bound of
Δ(x; ϕ). Some other related topics are also discussed.
On the Mean-Square for the Approximate Functional Equation of the Riemann Zeta-Function