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

Sandro Bettin, Alessandro Fazzari

We compute the one-level density of the non-trivial zeros of the Riemann zeta-function weighted by $|\zeta (\frac{1}{2}+it){|}^{2k}$ for $k=1$ and, for test functions with Fourier support in…