Recently Conrey, Farmer and Zirnbauer [CFZ1, CFZ2] conjec tur d formulas for the averages over a family of ratios of products of s hiftedL-functions. Their L-functions Ratios Conjecture predicts both the main and low er order terms for many problems, ranging fromn-level correlations and densities to mollifiers and moments to vanishing at the central point. There are now many results howing agreement between the main terms of number theory and random matrix theor y; h wever, there are very few… CONTINUE READING