Distribution modulo 1 of a linear sequence associated to a multiplicative function evaluated at polynomial arguments

Abstract

In two previous papers, the first named author jointly with Florian Luca and Henryk Iwaniec, have studied the distribution modulo 1 of sequences which have linear growth and are mean values of multiplicative functions on the set of all the integers. In this note, we give a first result concerning sequences with linear growth associated to the mean values of multiplicative functions on a set of polynomial values, proving the density modulo 1 of the sequence $$ \left( {\sum\limits_{m \leqslant n} {\frac{{\phi (m^2 + 1)}} {{m^2 + 1}}} } \right)_n . $$ This result is but an illustration of the theme which is currently being developed in the PhD thesis of the second named author.