On a mean value formula for the approximate functional equation of \(\zeta(s)\) in the critical strip. (English) Zbl 1176.11040

Feng, Shao-Ji

J. Math. Soc. Japan 57, No. 2, 513-521 (2005).

Summary: In a recent paper, I. Kiuchi and N. Yanagisawa [Arch. Math. 78, No. 5, 378–385 (2002; Zbl 1017.11046)] studied the even power moments of the error term in the approximate functional equation for \(\zeta(s)\). They got a mean value formula with an error term \(O(T^{1/2-k\sigma})\), and then they conjectured that this term could be replaced by \(E_{k,\sigma}T^{1/2-k\sigma}(1+o(1))\) 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 \(f(T)^{1/2-k\sigma}+o(T^{1/2-k\sigma})\) with \(f(T)\) oscillating.

MSC:

11M06

\(\zeta (s)\) and \(L(s, \chi)\)

Keywords:

even power moments; error term; approximate functional equation; Riemann zeta function; mean value formula

Citations:

Zbl 1017.11046