Premium
The sum‐of‐digits function of polynomial sequences
Author(s) -
Drmota Michael,
Mauduit Christian,
Rivat Joël
Publication year - 2011
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdr003
Subject(s) - mathematics , combinatorics , modulo , sequence (biology) , partition (number theory) , integer (computer science) , prime (order theory) , base (topology) , discrete mathematics , integer sequence , prime number , function (biology) , polynomial , arithmetic , generating function , computer science , mathematical analysis , genetics , evolutionary biology , biology , programming language
Let q ⩾2 be an integer and s q ( n ) denote the sum of the digits in base q of the positive integer n . The goal of this work is to study a problem of Gelfond concerning the re‐partition of the sequence ( s q ( P ( n ))) n ∈ℕ in arithmetic progressions when P ∈ℤ[ X ] is such that P (ℕ)⊂ℕ. We answer Gelfond's question and we show the uniform distribution modulo 1 of the sequence (α s q ( P ( n ))) n ∈ℕ for α∈ℝ \ ℚ, provided that q is a large enough prime number co‐prime with the leading coefficient of P .