Generalized Fourier coefficients of multiplicative functions

We introduce and analyze a general class of not necessarily bounded multiplicative functions, examples of which include the function n bar right arrow delta(omega(n)), where delta is an element of R \ {0} and where omega counts the number of distinct prime factors of n, as well as the function n bar right arrow vertical bar lambda(f)(n)vertical bar, where lambda(f)(n) denotes the Fourier coefficients of a primitive holomorphic cusp form. For this class of functions we show that after applying a W-trick, their elements become orthogonal to polynomial nilsequences. The resulting functions therefore have small uniformity norms of all orders by the Green-Tao-Ziegler inverse theorem, a consequence that will be used in a separate paper in order to asymptotically evaluate linear correlations of multiplicative functions from our class. Our result generalizes work of Green and Tao on the Mobius function.