×

The fourth moment of derivatives of the Riemann zeta-function. (English) Zbl 0644.10028

Let \(A_ i(s)=P_ i(-\frac 1L \frac d{ds})\zeta (s)\), \(1\leq i\leq 4\), where \(P_ i\) are polynomials and \(L=\log (\frac T{2\pi})\). The author proves that \[ \int^{T}_{1}A_ 1A_ 2(\tfrac 12+it)A_ 3A_ 4(\tfrac 12-it)\,dt \sim c(P_ 1,P_ 2,P_ 3,P_ 4)TL^ 4/(\tfrac{\pi^ 2}6). \] It follows that, for instance, \[ \int^{T}_{1}| \zeta '(+it)|^ 4\,dt \sim \frac{61}{1680\pi^ 2}TL^ 8. \] The proof is based on Y. Motohashi’s method [Proc. Japan. Acad., Ser. A 59, 393–396 (1983; Zbl 0541.10032)] to obtain approximate functional equations for the \(A_ i(s)\), and on A. E. Ingham’s method [Proc. Lond. Math. Soc. (2) 27, 273–300 (1928; JFM 53.0313.01)].

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
PDFBibTeX XMLCite
Full Text: DOI