Evaluations on the mean value formula of Atkinson-type for modular \(L\) functions (Q2858098)

The original Atkinson formula refers to the explicit expression for the function NEWLINE\[NEWLINE E(T) := \int_0^T|\zeta(1/2+it)|^2\,dt - T(\log(T/2\pi) + 2\gamma -1), NEWLINE\]NEWLINE where \(\gamma = -\Gamma'(1)\) is Euler's constant. This was obtained by \textit{F. V. Atkinson} [Acta Math. 81, 353--376 (1949; Zbl 0036.18603)] and sparkled much research on the mean square formulas for the Riemann zeta-function \(\zeta(s)\) and allied \(L\)-functions. The present author is interested in the evaluation of integrals involving the function NEWLINE\[NEWLINE Z_\varphi(t) := i^{-k/2}\psi^{-1/2}(k/2 + iy)\varphi(k/2+it),\leqno(1) NEWLINE\]NEWLINE which corresponds to \(Z^2(t)\), where NEWLINE\[NEWLINE Z(t) := \zeta(1/2+it)\chi^{-1/2}(1/2+it),\quad \zeta(s) = \chi(s)\zeta(1-s) NEWLINE\]NEWLINE is the classical Hardy function (see e.g., the reviewer's monograph [The theory of Hardy's \(Z\)-function. Cambridge: Cambridge University Press (2013; Zbl 1269.11075)]), well known from the investigation of the zeros of \(\zeta(s)\) on the ``critical line'' \(\Re s = 1/2\). In (1) \(\varphi(s)\) is the \(L\)-function attached to a holomorphic cusp form of weight \(k\) for the full modular group, which is an eigenfunction of all Hecke operators, with properly normalized coefficients. Its functional equation is NEWLINE\[NEWLINE \varphi(s) = (-1)^{k/2}\psi(s)\varphi(k-s), \quad \psi(s) = (2\pi)^{2s-k} {\Gamma(k-s)\over \Gamma(s)}, NEWLINE\]NEWLINE analogous to the functional equation for \(\zeta^2(s)\). \(Z_\varphi(t)\) is a real-valued, oscillating function, and \textit{M. Jutila} [in: Voronoï's impact on modern science. Book I. Translated from the Ukrainian. Kyiv: Institute of Mathematics, 137--154 (1998; Zbl 0948.11032)] obtained an analogue of Atkinson's formula for \(E(T)\) for the function NEWLINE\[NEWLINE E_\varphi(T) := \int_0^T Z_\varphi(t)\,dt.\leqno(2) NEWLINE\]NEWLINE The author uses Jutila's expression for (2), together with ideas of earlier works of \textit{D. R. Heath-Brown} [Mathematika 25, 177--184 (1979; Zbl 0387.10023)] and \textit{D. R. Heath-Brown} and \textit{K. M. Tsang} [J. Number Theory 49, No. 1, 73--83 (1994; Zbl 0810.11046)] to obtain a mean square formula for \(E_\varphi(T)\), namely NEWLINE\[NEWLINE \int_0^T E_\varphi^2(t)\,dt = CT^{3/2} + O(T^{5/4}\log^2T),\quad C = C_\varphi > 0.\leqno(3) NEWLINE\]NEWLINE He also shows that there exist two positive constants \(A,B\) such that every interval \([T, T + A\sqrt{T}]\, (T\geq T_0), \) contains two points \(t_1, t_2\) such that NEWLINE\[NEWLINE E_\varphi(t_1) \geq Bt_1^{1/4}, \quad E_\varphi(t_2) \leq -Bt_2^{1/4}. \leqno(4) NEWLINE\]NEWLINE Reviewers remark:NEWLINENEWLINE 1. The omega results (4) follow also, in a somewhat different way, from the reviewer's paper [Acta Arith. 56, No. 2, 135--159 (1990; Zbl 0659.10053)], published before the work of Heath-Brown and Tsang. \medskip 2. A further improvement of the mean square formula for \(E_\varphi(T)\) will result from the use of an analogue of \textit{T. Meurman}'s formula for \(E(T)\) [Q. J. Math., Oxf. II. Ser. 38, 337--343 (1987; Zbl 0624.10032)], which enabled Meurman to improve on Heath-Brown's mean square formula for \(E(T)\) and to obtain NEWLINE\[NEWLINE \int_0^T E^2(t)\,dt = cT^{3/2} + O(T\log^5T),\quad c>0. NEWLINE\]NEWLINE A similar improvement concerning (3) is possible.