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Multiplicative functions dictated by Artin symbols

Robert J. Lemke Oliver — 2013

Acta Arithmetica

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 $_{K}$ of completely multiplicative functions whose values are dictated by Artin symbols, and we show that...

Almost-primes represented by quadratic polynomials

Robert J. Lemke Oliver — 2012

Acta Arithmetica

On the counting function for the generalized Niven numbers

Ryan Daileda; Jessica Jou; Robert Lemke-Oliver; Elizabeth Rossolimo; Enrique Treviño — 2009

Journal de Théorie des Nombres de Bordeaux

Given an integer base $q \geq 2$ and a completely $q$ -additive arithmetic function $f$ taking integer values, we deduce an asymptotic expression for the counting function $N_{f} (x) = # \{0 \leq n < x | f (n) ∣ n\}$ under a mild restriction on the values of $f$ . When $f = s_{q}$ , the base $q$ sum of digits function, the integers counted by $N_{f}$ are the so-called base $q$ Niven numbers, and our result provides a generalization of the asymptotic known in that case.

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Multiplicative functions dictated by Artin symbols

Almost-primes represented by quadratic polynomials

On the counting function for the generalized Niven numbers