Multiplicative Functions Dictated by Artin Symbols

Granville and Soundararajan have recently suggested that a general study of multiplicative functions could form the basis of analytic number theory without zeros of L-functions; this is the so-called pretentious view of analytic number theory. Here we study multiplicative functions which arise from the arithmetic of number fields. For each finite Galois extension K/Q, we construct a natural class SK of completely multiplicative functions whose values are dictated by Artin symbols, and we show that the only functions in SK whose partial sums exhibit greater than expected cancellation are Dirichlet characters.