On some mean value results involving \(| \zeta(1/2+it)| \). (English) Zbl 1055.11051

Ivić, A.

Bull., Cl. Sci. Math. Nat., Sci. Math. 127, No. 28, 17-29 (2003).

Let \(E_1(T)\) and \(E_2(T)\) denote the error terms in the asymptotic formula for the second and for the fourth moment of the modulus of the Riemann zeta-function on the critical line, respectively, i.e., for \(k=1,2\), \[ E_k(T)=\int_0^T| \zeta(1/2+it)| ^{2k}\,dt-TP_{2k}(\log T), \] where \(P_{2k}\) is a polynomial of degree \(2k\). In this nicely written paper it is proved that \[ I(T):=\int_T^{2T}E_1(t)E_2(t)\,dt\ll T^{7/4}(\log T)^{7/2}\log\log T. \] The method of proof relies on an explicit formula due to A. Ivić and Y. Motohashi [J. Number Theory 51, 16–45 (1995; Zbl 0824.11048)] and on the relation \[ \int_T^{2T}E_1(t)E_2(t)\,dt =-\int_T^{2T}t^{3/4}g(t)| \zeta(1/2+it)| ^4\,dt+O(T^{3/2}(\log T)^{10}), \] where \(g(t)\) is given by \[ g(t)={1\over2}\left({2\over\pi}\right)^{3/4}\sum_{n=1}^\infty (-1)^nd(n)n^{-5/4}\sin(\sqrt{8\pi nt}-\pi/4). \] Denote by \(g_+(t)\) and \(g_-(t)\) the maximum and the minimum of \(g(t)\) and the function constant zero, respectively. Then, it is further shown that, for \(k=1,2\), \[ \int_T^{2T}t^{3/4}g_{\pm}(t)| \zeta(1/2+it)| ^{2k}\,dt \asymp T^{7/4}(\log T)^{k^2}. \] Comparing this with the other estimate from above it follows that there must be some cancellation for \(g(t)=g_+(t)+g_-(t)\). Moreover, it is proved that \[ \int_T^{2T}t^{3/4}g(t)| \zeta(1/2+it)| ^2\,dt=-BT^{3/2}+O(T^{5/4}\log T), \] which makes it reasonable to think that also \(I(T)\) is of the order \(T^{3/2+o(1)}\).

Reviewer: Jörn Steuding (Madrid)

MSC:

11M06

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

Keywords:

Riemann zeta-function

Citations:

Zbl 0824.11048