On the Correlation of Shifted Values of the Riemann Zeta Function

In 2007, assuming the Riemann hypothesis, Soundararajan (Moments of the Riemann zeta-function, Ann. Math., 170 (2009) 981–993) proved that ∫0T|ζ(½ + it)|2k dt ≪k, ϵT(log T)k2 + ϵ for every positive real number k and every ϵ>0. In this paper, we generalize his methods to find upper bounds for shifted moments. We also obtained their lower bounds and conjectured asymptotic formulae based on the random matrix model, which is analogous to Keating and Snaith's work. These upper and lower bounds suggest that the correlation of |ζ(½ + it + iα1)| and |ζ(½ + it + iα2)| transition at |α1−α2| ≈ 1/log T. In particular, these distributions appear to be independent when |α1 − α2| is much larger than 1/log T.