Nonlinear twists and moments of \(L\)-functions (Q6595580)

Let \(F\) be a function in the extended Selberg class \(S^\sharp\) of integer degree \(d \geq 1\) with \(\theta_F = 0\). Let \(\delta\geq 0\). it is shown that if the self-reciprocal twists \(F^2_{\mathrm{self}}(s)\) and \(\overline{F^2}_{\mathrm{self}}(s)\) have holomorphic continuation to the half-plane \(\sigma> \frac{1}{2}+\frac{1}{2d}+\delta\) with polynomial growth on every vertical strip contained in this half-plane (\(\delta\)-Hypothesis), then for any \(\varepsilon>0\) we have \N\[\N\int_{-T}^T |F(1/2+it)|^2 dt\ll T^{1+\delta d+\varepsilon}.\N\]\NAs consequence, if \(F(s)=\zeta(s)^k\) with integer \(k\geq 1\), then a sharp bound for the mean-square of \(\zeta(1/2+it)\) is derived conditionally under the \(\delta\)-Hypothesis.