Uniform Distribution of Sequences of Integers in Residue Classes
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1087 Wtadystaw Narkiewicz
Uniform Distribution of Sequences of Integers in Residue Classes
Spri nger-Verlag Berlin Heidelberg New York Tokyo 1984
Author
Wtadystaw Narkiewicz Wroclaw University, Department of Mathematics Plac Grunwaldzki 2-4, 50-384 Wrocfaw, Poland
AMS Subject Classification (1980): lOA35, 10023, lOH20, 10H25, 10L20, lOM05 ISBN 3-540-13872-2 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13872-2 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach / Bergstr. 2146/3140-543210
To my teacher Professor
Stanis):aw Hartman
on his seventieth anniversary
INTRODUCTION
The aim of these notes, which form an extended version of lectures given by the author at various places, is to present a survey of what is known about uniform distribution of sequences of integers in residue classes. Such sequences were studied since the beginning of this century, when L.E.Dickson in his Ph.D. thesis made a thorough study of permutational polynomials, i.e. polynomials inducing a permutation of residue classes with respect to a fixed prime. We shall also consider weak uniform distribution of sequences, meaning by that uniform distribution in residue classes (mod N), prime to N. The standard example here is the sequence of all primes, which is weakly uniformly distributed (mod N) for every integer
N.
After proving, in the first chapter, certain general results we shall consider uniform distribution of certain types of sequences, starting with polynomial sequences and considering also linear recurrent sequences and sequences defined by values of additive arithmetical functions. This will be done in chapter IIIV. In the last two chapter we shall study uniform distribution of sequences defined by multiplicative functions,in particular those, which are "polynomiallike", i.e. satisfy the condition polynomials
f(pk)
= Pk(p)
for primes
p and
k
with suitable
P 1,P2, . . . . In particular we shall consider classical
arithmetical functions, like the number or sum of divisors, Euler's wfunction and Ramanujan's
Tfunction. This will lead to certain ques-
tions concerning the value distribution of polynomials. Our tools'belong to the classical number theory and include fundamentals of the theory of algebraic numbers. In certain places we shall use more recent work, like the theorems of p.Deligne, J.P.Serre
and
H.P.F.SwinnertonDyer on modular forms, which will be used in the study of Ramanujan's function. In such cases we shall explicitly state the result needed wit
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