A weighted one-level density of the non-trivial zeros of the Riemann zeta-function

We compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by \(|\zeta (\frac{1}{2}+it)|^{2k}\) $|\zeta (\frac{1}{2}+it){|}^{2k}$ for \(k=1\) $k=1$ and, for test functions with Fourier support in \((-\frac{1}{2},\frac{1}{2})\) $(-\frac{1}{2},\frac{1}{2})$ , for \(k=2\) $k=2$ . As a consequence, for \(k=1,2\) $k=1,2$ , we deduce under the Riemann hypothesis that \(T(\log T)^{1-k^2+o(1)}\) $T{(logT)}^{1-k^2+o(1)}$ non-trivial zeros of \(\zeta \) $\zeta$ , of imaginary parts up to T, are such that \(\zeta \) $\zeta$ attains a value of size \((\log T)^{k+o(1)}\) ${(logT)}^{k+o(1)}$ at a point which is within \(O(1/\log T)\) $O(1/logT)$ from the zero.