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Multiplicative functions dictated by Artin symbols

Volume 161 / 2013

Robert J. Lemke Oliver 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.

Authors

  • Robert J. Lemke OliverDepartment 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.
    e-mail
    e-mail

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