On a mean value formula for the approximate functional equation of $\bm{\zeta(s)}$ 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 $\zeta \left(s\right)$. They got a mean value formula with an error term $O\left(T^{1/2-k\sigma }\right)$, and then they conjecture that this term could be replaced by $E_{k,\sigma }{T}^{1/2-k\sigma }(1+o(1\left)\right)$ with constant $E_{k,\sigma }$ depending on $k$ and $\sigma$. In this paper, we disprove this conjecture by showing that the error term should be $f\left(T\right)T^{1/2-k\sigma }+o\left(T^{1/2-k\sigma }\right)$ with $f\left(T\right)$ oscillating.