On a mean value formula for the approximate functional equation of \bm{ξ(s)} in the critical strip

Abstract

In a recent paper, Isao Kiuchi and Naoki Yanagisawa studied the even power moments of the error term in the approximate functional equation for ξ(s). They got a mean value formula with an error term O(T^1/2-kσ), and then they conjecture that this term could be replaced by E_{k, σ}T^1/2-kσ(1+o(1)) with constant E_{k, σ} depending on k and σ. In this paper, we disprove this conjecture by showing that the error term should be f(T)T^1/2-kσ+o(T^1/2-kσ) with f(T) oscillating.