ON THE CORRELATION OF SHIFTED VALUES OF THE RIEMANN ZETA FUNCTION

Abstract

In 2007, assuming the Riemann Hypothesis (RH), Soundararajan [11] proved that R T 0 |ζ(1/2 + it)|2k dt ≪k,ǫ T(log T) k2 +ǫ for every k positive real number and every ǫ> 0. In this paper I generalized his methods to find upper bounds for shifted moments. We also obtained their lower bounds and conjectured 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 |ζ ( 1 2 + it+iα1) | and |ζ ( 1 2 + it+iα2) | transition at |α1 −α2 | ≈ 1 log T |α1 − α2 | is much larger than. In particular these distribution appear independent when 1/log T.