Moments of Random Multiplicative Functions and Truncated Characteristic Polynomials

Abstract

An asymptotic formula for the 2kth moment of a sum of multiplicative Steinahus variables is given. This is obtained by expressing the moment as a 2k-fold complex contour integral, from which one can extract the lead- ing order term. The 2kth moment of a truncated characteristic polynomial of a unitary matrix is also computed. This is done by expressing the moment as a combinatoric sum over a restricted region, and then invoking each restric- tion by introducing some complex integral. This gives a 2k-fold integral that is very similar to the 2kth moment of the sum of multiplicative Steinhaus variables, which in turn gives an asymptotic relation between the two. Similarly, an asymptotic formula is given for the 2kth moment of a sum of multiplicative Rademacher variables, and the 2kth moment of the truncated characteristic polynomial of a special orthogonal matrix is found. This gives an asymptotic relation between these two.