WENTS Seminar: On the correlation of shifted values of the Riemann zeta function

In 2007, assuming the Riemann Hypothesis (RH), Soundararajan \cite{Moment} proved that the 2k-th moments of the Riemann zeta function on the critical line is bounded by T(log T)^{k^2 + epsilon} for every k positive real number and every epsilon > 0. In this talk I will generalize his methods to find upper bounds for shifted moments. Also I will sketch the proof how we derive their lower bounds and conjecture asymptotic formulas based on Random matrix model, which is analogous to Keating and Snaith's work. These upper and lower bounds suggest that the correlation of |\zeta(1/2 + it + i\alpha_1)| and |\zeta(1/2 + it + i\alpha_2)| transition at |\alpha_1 - \alpha_2| is around 1/log T}. In particular these distribution appear independent when |\alpha_1 - \alpha_2| is much larger than 1/log T.