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 |ζ(12+it)|2k for k=1 and, for test functions with Fourier support in (-12,12), for k=2. As a consequence, for k=1,2, we deduce under the Riemann hypothesis that T(logT)javax.xml.bind.JAXBElement@2aba6848javax.xml.bind.JAXBElement@4abf1d3bkjavax.xml.bind.JAXBElement@5508e36djavax.xml.bind.JAXBElement@2bd394ebjavax.xml.bind.JAXBElement@52d251f1javax.xml.bind.JAXBElement@6036c855 non-trivial zeros of ζ, of imaginary parts up to T, are such that ζ attains a value of size (logT)k+o(1) at a point which is within O(1/logT) from the zero.