## 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 $L$-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 $K/\mathbb {Q}$, 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.