On a mean value formula for the approximate functional equation of ζ(s) in the critical strip

摘要

In a recent paper, Isao Kiuchi and Naoki Yanagisawa studied the even power moments of the error term in the approximate functional equation for ζ(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(l)) 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.