Number Theory: A Seminar held at the Graduate School and University Center of the City University of New York 1982

313 Pages·1984·4.663 MB·English

by  Krishnaswami Alladi (auth.), David V. Chudnovsky, Gregory V. Chudnovsky, Harvey Cohn, Melvin B. Nathanson (eds.)

Lecture Notes ni
Mathematics
Edited by .A Dold dna .B nnamkcE
1052
rebmuN yroehT
A Seminar held School the Graduate at dna
Center University of the City University of
NewYork 1982
Edited yb .V.D G.V. Chudnovsky, ,yksvonduhC
.H Cohn and .M .B Nathanson
I I
galreV-regnirpS
nilreB Heidelberg New York oykoT 1984
srotidE
David .V Chudnovsky
Gregory .V Chudnovsky
Department of Mathematics, Columbia University
NewYork, NY 10027, USA
Harvey Cohn
Department of Mathematics, .Y.N.U.C City College
NewYork, NY ,13001 USA
Melvyn .B Nathanson
Department of Mathematics, Rutgers - The State University
Newark, JN 07102, USA
AMS Subject Classification 10-06 (1980):
ISBN 3-54042909-X Springer-Verlag Berlin Heidelberg New York oykoT
ISBN 0-38742909-X Springer-Verlag New York Heidelberg Berlin Tokyo
Library of Congress Cataloging in Publication Data. Main entry under title: Number theory.
(Lecture notes in mathematics; 1052) .1 Numbers, Theory of-Addresses, essays, lectures.
.I Chudnovsky, D. (David), 1947-. .II NewYork Number Theory Seminar (1982-1983) III. Series:
Lecture notes in mathematics (Springer-Verlag); 1052.
QA3.L28 no. 1052 [QA241] 510s [512'.7] 84-1360
ISBN 0-387q2909-X (U.S.)
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INTRODUCT ION
In January, 1982 four number theorists - David and Gregory
Chudnovsky, Harvey Cohn, and Melvyn B. Nathanson - organized the
New York Number Theory Seminar. The Seminar met weekly during the
spring, 1982 semester at the Graduate School and University Center
of the City University of New York at II West 42 Street in
Manhattan. This volume contains expanded texts of the lectures
delivered in the Seminar. The Seminar continued in the 1982-83
academic year, and the reports presented in this second year will
be published in a subsequent volume.
The organizers hope that the New York Number Theory
Seminar will provide a continuing opportunity to discuss recent
results in the higher arithmetic, and that the publication of
the annual proceedings will contribute to research in number
theory.
TABLE OF CONTENTS
K. ALLADI , Moments of Additive Functions and Sieve
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . .
H. COHEN and H.W. LENSTRA, JR. , Heuristics on Class
Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
D.V. CHUDNOVSKY and G.V. CHUDNOVSKY , Pad~ and Rational
Approximations to Systems of Functions and their Arithmetic
Applications . . . . . . . . . . . . . . . . . . . . . . . . . 37
D.V. CHUDNOVSKY and G.V. CHUDNOVSKY , Pad~ Approximations
to Solutions of Linear Differential Equations and Applications
to Diophantine Analysis . . . . . . . . . . . . . . . . . . . . 85
J. DIAMOND , p- adic Gamma Functions and their Applications . . 168
J.C. LAGARIAS and A.M. ODLYZKO , New Algorithms for
Computing ~(x) . . . . . . . . . . . . . . . . . . . . . . . . 176
J. LEPOWSKY and M. PRIMC , Standard Modules for Type One
Affine Lie Algebras . . . . . . . . . . . . . . . . . . . . . . 194
C. MORENO , Some Problems of Effectivity in Arithmetic,
Geometry and Analysis . . . . . . . . . . . . . . . . . . . . . 252
M.B. NATHASON , The Exact Order of Subsets of Additive
Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
C.F. OSGOOD , Effective Bounds on the Diophantine Approxi-
mations of Algebraic Functions, and Nevanlinna Theory ..... 278
P. SARNAK , Additive Number Theory and Maass Forms . . . . . . 286
Moments of additive functions and sieve methods
Krishnaswami Alladi
il. Introduction
We shall report here our recent work on the estimation of moments of additive
functions f(n), for values n restricted to certain subsets S of the positive
integers. Although of relatively recent origin, additive functions have been an
object of intense study in the past few decades because a number of impressive dis-
tribution results could be established using a variety of techniques. Most research
however relates to the set ~+ of all positive integers and the problem concerning
subsets S of ~+ has received little attention. The purpose of this exposition
is to describe a new method that we employ, which enables us, amongst other things,
to extend various classical results to certain subsets S; besides, our method may
have other implications as well. As far as we know, it is the first occasion when
the sieve method has been used in such moment problems and this results in a satis-
factory treatment of a wide class of sets S.
So far the main interest in employing the moment method to study the distri-
bution of additive functions has been due to its entirely elementary nature, but
this in turn led to several tedious calculations. Our approach eliminates much of
this difficulty by using the machinery of bilateral Laplace transforms.
Our paper is basically divided into two parts. Up to S4 we discuss several
classical results and remark on their merits and limitations. From 55 to 510 we
describe our method and results and compare these with earlier approaches. We state
our results in 558 and .9 Finally in §i0 we briefly discuss limitations in our tech-
nique and indicate directions for further work and progress.
Our method is an improvement of a recent technique due to Elliott [6] who ob-
tained uniform upper bounds for the moments of arbitrary additive functions, in the
situation S = ~+. Later, in 59, we shall point out similarities and differences
between Elliott's approach and ours.
We only discuss the main ideas here and not give details of proofs. A more
complete treatment of our method can be found in [i]. For now, we conclude this
section by collecting some notations and conventions.
Recall that an additive function f(n) is an arithmetical function that satis-
fies f(mn) = f(m) + f(n) for integers m,n with g.e.d. (m,n) I. = Similarly a
Talk given at the New York Number Theory Seminar on April 12, 1982.
multiplicative function g satisfies g(mn) = g(m)g(n) for = (m,n) .i Thus additive
and multiplicative functions are completely determined by their values on prime powers
pe, e e ZZ . + For simplicity we concentrate here on strongly additive functions f
which satisfy
f(n) = ~ f(p) . (I.i)
Pln
p=prime
Similarly a strongly multiplicative function g is given by
g(n) = R g(p) . (1.2)
Pln
As usual emply sums equal zero and empty products, one.
The sets S we discuss will satisfy some conditions imposed upon the quantity
Sd(X) = ~ n , a (i. 3)
,x__<n ne S
O---n door( d)
where [a } n is a sequence of 'weights' attached to S. The weights are ~ 0 and
we write
X~ )d(
Sd(X) = d + Rd(X) ' (I .4)
where X = Sl(X). We require that
)d(~c is multiplicative, > 0 and c~(p) is bounded . (1.5)
In addition we need
"average of IRd(X) I is in some sense small" .i( 6)
We shall make (1.6) more precise in the sequel.
With S as above we associate with each strongly additive ,f the sums
A(x) = Af(x)= ~ f(p)co(p) (1,7)
x_<p P
and
Bk(X) = .~ l if(p) , w(p) k for k > 0 . (1.8)
x_<p P
As usual the << and '0' notation are equivalent and will be used inter-
changeably as is convenient. Unless indicated otherwise, implicit constants are
absolute, or depend at most upon S and this will be clear from the context. The
Moebius function g(n) is the multiplicative function given by ~(p) = -I for each
prime p, and ~(pe) 0 = for all p and e ~ 2. We denote by p(n) the smallest
prime factor of n if n > ,I and p(1) =~. Finally P(y) = ~ p. Further notation
will be introduced when needed. Y-<P
52. Classical distribution results
Despite the fact that prime numbers have been the focus of attention in Number
Theory since Greek antiquity, the first significant results on v(n), the number of
prime divisors of n, were established only in the twentieth century. More pre-
cisely Hardy and Ramanujan [13] observed in 1917 that v(n) is almost always nearly
log logn in size. This was really the starting point for the study of additive
functions although the subject took shape only two decades later.
Intensive study began in 1934 when P. Turan [19] showed that
(f(n) - A(x))2~f xA(x) (2.1)
x_<n
holds for strongly additive functions f satisfying 0 ~ f(p) << I. In particular
from (2.1) with f(n) = v(n), the Hardy-RamanuJan result follows~ But more impor-
tantly, for the first time, the role of probability theory in the study of additive
functions could be perceived because (2.1) is essentially an estimate for the second
moment (variance) of .f
This led Tur~n's Hungarian colleague Erd~s to an active study of the mean values
of additive functions [7; ,I II, IIII. The efforts of Erd~s culminated in two major
theorms which clearly set the study of additive functions on a firm probabilistic
foundation. One of these established jointly with Wintner [9], provided necessary
and sufficient conditions for the frequencies
x
n<x
f (n)!v
of an additive function to converge weakly (in v) as x + ~. The second, established
jointly with Kac [8], considered the value f(n), i < n < x, in terms of sums of
nearly independent random variables, one for each prime p ~ x. When compared to a
sum of independent random variables, it showed under the influence of the Central
Limit Theorem that
aril F (v) = G(v) = ~ I S v e-U2/2 du (2.2)
x
for all real strongly additive f for which
f(p) = 0(I) , and B2(x ) + with x , (2.3)
where
I
Fx(V) = ~ ~ i (2.4)
n<x
A-)n~f (x)~ fB )x(
(In establishing (2.2) they made crucial use of a certain sieve estimate due to
Brun; further light on this will be shed in S3.) Thus the study of additive
functions (alternatively Probabilistic Number Theory), gained momentum.
For additive functions taking certain simple values on primes, R~nyi and Turin
[18] demonstrated that an alternate elegant treatment is possible. More precisely
iu
with e = z , u real, they noticed that the sum of multiplicative functions
z f(n) (2.5)
n<x
could be estimated asymptotically by familiar analytic techniques. The sum in (2.5)
is related to the characteristic function (Fourier transform) of the distribution
Fx(V ) and hence leads to the determination of the limit of Fx(V) as x + ~. In
particular for v(n) they used (2.5) to explicitly calculate the rate of converg-
ence in (2.2).
The next major advance was made by Kubilius in 1956 [15], who successfully com-
bined all of the above ideas (see also [16] for a more elaborate treatment). By
employing several tools from probability theory such as infinitely divisible distri-
butions, independent random variables and characteristic functions, he obtained
necessary and sufficient conditions for the frequencies Fx(V) in (2.4) to con-
verge weakly (in v) as x + ~, for all functions of a certain class ~. This
class comprised of all strongly additive functions for which B2(x) + ~ with x
and
2 B (x) ~ + ~ .
B2(Y ) + i as x ~ ~ for some ~ = ~(x) = log y (2.6)
He produced the first major examples of additive functions f with the limit in
(2.2) different from the Gaussian distribution. Kuhilius was able to determine
the limit ~(v) of the characteristic functions ~x(v) of Fx(V) and thus, in
principle, determined the weak limit of F
x
In a somewhat different direction, it was realised soon after Tura~ established
(2.1), that it would be useful to generalise it to arbitrary strongly additive
functions. This was done by Kubilius in 1956 who showed that
f(n)'A(x) I 21 << xB 2(x) )7.2(
n<x
holds uniformly for all such .f Special cases of (2.6) were used by Erd~s-Kac,
Erd~s-Wintner and Kubilius to establish their distribution results. There will be
more on this in the sequel.
.3$ Brun's sieve in Probabilistic Number Theory
First we sketch the ideas due to Erd~s-Kac.
For each prime p define a function
)n(p0 I I if p ln
=
0 if p~n .
Then every strongly additive function can be written sa
f(n) = ~ f(p)Op(n) .
P
The functions pO have the property that in the interval [l,x]
I I with probability ~ i/p
0p(n) = i ' for p = o(x) .
0 with probability ~(i ~) -
Also, for p=o(x), the ) oD(n are nearly independent .ni [l,x]. The idea si to
compare f(n) with the sum
X=~x
P
P
of infinite independent random variables defined by
I f(p) with probability I/p
Xp = 0 with probability - I _I .
P
If B2(Y) + ~ with ,y it follows that the random variable
Xy = Z Y_<P ~ P O.1)
has mean and variance asymptotically equal to A(y) and ~ respectively.
Furthermore if O = f(p) (I), then by the Central Limit Theorem the random variable
~ / } (Xy-A(y) )y( )2.3(
has a distribution function --+ G(v) as y + ~.
It is only natural to compare the quantity in (3.2) with
[fy(n)-A~)}/~--~ , (3.3)
where f (n) is the truncated additive function
Y
f (n) = ~ f(p) . (3,4)
Y pln
y_<p
While comparing (3.2) and (3.3), Erd~s and Kac required an estimate for the quantity
~(x,y) = ~ i
n<x
p (n)>y
On a probabilistic basis one expects that
(x,y) ~ x n (I-~) , as x +~ . (3.5)
P<J
In fact they observed that the random variables in (3.2) and (3.3) have asymptotic-
ally the same distribution function (the Gaussian) if (3.5) is true. Burn's sieve
method shows that (3.5) holds so long as
= log x/log y -~ ~ with x .
(In fact (3.5) is false if ~-+ ~ with ).'x
To complete the transition from f (n) to f(n) they used Turan's inequality
Y __
(2.1) to deduce that for the function f= f-f
Y
[~(n)-A_(x)} 2 = o(xB2(x)) (3.6)
x__<n f
holds for a suitable choice of y and ~ + ~, on the basis of (2.3).
Although it was implicit in the work of Erd~s-Kac that their method applies
more generally to sets S 7z+ c where the analogue of Brun's estimate (3.5) holds,
this was carried out only by Kubilius some years later. In doing so Kubilius used
the method of characteristic functions and thus avoided the use of the Central
Limit Theorem. This enabled him, amongst other things, to consider additive
functions of growth faster than in (2.3) and therefore could obtain significant re-
sults with limiting distributions other than the Gaussian.
Consider a set S for which (1.4) and (1.5) hold. Assume also the following
precise version of (1.6): For each b there exists c such that
X (3.7)
I IRd(X) ~b logbx "
d<_X/logex