Multiplicative functions dictated by Artin symbols

Robert J. Lemke Oliver¹

¹ Department of Mathematics and Computer Science Emory University 400 Dowman Dr. Atlanta, GA 30322, U.S.A. and Department of Mathematics Stanford University Building 380 Stanford, CA 94305, U.S.A.

Acta Arithmetica, Tome 161 (2013) no. 1, pp. 21-31.

Résumé

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.

Détail

DOI : 10.4064/aa161-1-2

Keywords: granville soundararajan have recently suggested general study multiplicative functions could form basis analytic number theory without zeros l functions so called pretentious view analytic number theory here study multiplicative functions which arise arithmetic number fields each finite galois extension mathbb construct natural class mathcal completely multiplicative functions whose values dictated artin symbols only functions mathcal whose partial sums exhibit greater expected cancellation dirichlet characters