Multiplicative functions dictated by Artin symbols
Volume 161 / 2013
Acta Arithmetica 161 (2013), 21-31
MSC: Primary 11R42; Secondary 11M41.
DOI: 10.4064/aa161-1-2
Abstract
Granville and Soundararajan have recently suggested that a general study of multiplicative functions could form the basis of analytic number theory without zeros of -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/\mathbb {Q}, we construct a natural class \mathcal {S}_K of completely multiplicative functions whose values are dictated by Artin symbols, and we show that the only functions in \mathcal {S}_K whose partial sums exhibit greater than expected cancellation are Dirichlet characters.
