Asymptotics of the fourth power moment of the Riemann zeta-function in the critical strip (Q1366373)

This paper is the continuation of the author's previous work on the mean square of \(|\zeta(\sigma+it)|\) [\textit{A. Kačėnas}, Lith. Math. J. 35, 249-261 (1995); translation from Liet. Mat. Rink. 35, 315-331 (1995; Zbl 0861.11051)]. Now the author is concerned with the estimation of the function \[ \begin{aligned} E_{2,\sigma}(T) :&=\int_0^T|\zeta(\sigma+it)|^4 dt-{\zeta^4(2\sigma)\over \zeta(4\sigma)}T \\ &- \left(\sum_{j=0}^3a_j(\sigma)\log^{3-j}T\right)T^{2-2\sigma}-A(\sigma)T^{3-4\sigma}\end{aligned}\tag{1} \] for \(1/2 <\sigma<1\) (\(a_j(\sigma)\) and \(A(\sigma)\) are constants dependent on \(\sigma\), which may be evaluated explicitly, \(A(\sigma)=0\) for \(\sigma>3/4\)), which may be considered as the error term in the asymptotic formula for the biquadrate of \(|\zeta(\sigma+it)|\). The result of Theorem 1 may be formulated as \[ E_{2,\sigma}(T)\ll\begin{cases} T^{{75-56\sigma\over61}+\varepsilon} \quad &({1\over2} < \sigma<{229\over306}),\\ T^{{83\over153}+\varepsilon} \quad &({229\over306}\leq\sigma<1),\end{cases} \tag{2} \] where \(\varepsilon > 0\) is any given number. It may be remarked that \({83\over153}>2-2\sigma\) (the exponent in the second main term in (1)) for \(\sigma>{223\over306} = 0.72875\ldots \). The basic idea of proof is due to \textit{D. R. Heath-Brown} [Proc. Lond. Math. Soc., III. Ser. 38, 385-422 (1979; Zbl 0403.10018)], who employed an averaging technique (based on the use of the Gaussian exponential weight) coupled with results on the binary additive divisor problem (estimation of the error term in the asymptotic formula for \(\sum_{n\leq x}d(n)d(n+h)\), where \(d(n)\) is the number of divisors of \(n\) and \(h\) is not fixed), to obtain the bound \(E_{2,1/2}(T)\ll T^{7/8+\varepsilon}\). The advent of powerful methods from the spectral theory of the non-Euclidean Laplacian led to dramatic improvements in the fourth moment problem on the critical line. Thus \textit{Y. Motohashi} [Acta Math. 170, 181-220 (1993; Zbl 0784.11042)] proved an asymptotic formula for \(\int_{-\infty}^\infty|\zeta(1/2+it+iT)|^4e^{-(t/G)^2} dt (T,G > 0)\), which was used by him and \textit{A. Ivić} [J. Number Theory 51, 16-45 (1995; Zbl 0824.11048)] to obtain \(E_{2,1/2}(T) \ll T^{2/3}\log^CT\). The author also used the works of \textit{Y. Motohashi} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 27, 529-572 (1994; Zbl 0819.11038)] and \textit{Y. Motohashi} and the reviewer [Proc. Lond. Math. Soc., III. Ser. 69, 309-329 (1994; Zbl 0805.11060)] on the binary additive divisor problem in his proof of (1). However, it is possible to use the spectral-theoretic approach globally in dealing with \(E_{2,\sigma}(T)\) for \(1/2 < \sigma < 1\), as was done by the reviewer [J. Théor. Nombres Bordx. 8, 101-123 (1996; Zbl 0858.11045)], of whose work the author was unaware. The reviewer showed [in his def. (15) of \(E_{2,\sigma}(T)\) on p. 108 the terms \(j=2,3\) in (1) above are missing] that, for fixed \(1/2<\sigma<1\), \[ \int_0^T|\zeta(\sigma + it)|^4 dt = {\zeta^4(2\sigma)\over\zeta(4\sigma)}T + O\left(T^{2-2\sigma}\log^3T\right) \] and \(E_{2,\sigma}(T) \ll T^{2\over1+4\sigma}\log^CT.\) These results supersede the author's bounds in (2). In Theorem 2 the author is concerned with the asymptotic evaluation of \[ I_T :=\int_0^T|\zeta(\sigma_T+it)|^4 dt,\quad (\sigma_T ={1\over2}+{1\over\ell_T}), \] where \(\ell_T\) is a positive function tending to infinity with \(T\) (the definition of \(\sigma_T\) is missing in the paper). He obtains the following precise results, which elucidate the behaviour of the fourth moment integral near the critical line: \[ I_T \sim \begin{cases} {3\over8\pi^2}T\ell_T^4 &(\ell_T=o(\log T), T \to \infty)\\ {1\over2\pi^2}T\log^4T &(\log T = o(\ell_T), T \to \infty)\\ {3\over8\pi^2\kappa^4}(1-e^{-2\kappa})^2T\log^4T &(\lim_{T\to\infty}{\log T\over\ell_T} = \kappa > 0).\end{cases}. \]