Probability and Number Theory: Some Examples of Connections
JeanMarc Deshouillers
Mathématiques Stochastiques, Université Bordeaux 2
Algorithms Seminar
January 13, 1997
[summary by Alain Plagne]
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Abstract
We illustrate some connections between
probability theory and number theory. Examples are taken from multiplicative number theory
(probabilistic behaviour of the function w), additive number theory (Sidon problem,
where arithmetic results are proved by using probability
theory; partitions; and ErdösRényi type models
for sums of powers, where situations on which nothing is known, are modelled) and
probability (upper bound for the concentration of the sum of independent and identically
distributed integer random variables, where probabilistic results are obtained by
using methods from additive number theory).
1 Introduction
One can distinguish at least four types of connections between probability
theory and additive number theory.

(i) When probability leads to models for the integers;
 (ii) When probability techniques permit to prove number theory results;
 (iii) When number theory questions lead to probability questions;
 (iv) When number theory methods permit to prove probability results.
Part 2 is devoted to multiplicative number theory. It provides
examples for points (i) and (ii).
The third part, which is the heart of this talk, is concerned with additive
number theory and illustrated with Sidon's problem (point (ii)), partitions (points
(i) and (ii)) and sums of sth powers (points (i) and (iii)).
In the fourth part, which deals with concentration functions, point (iv) is
illustrated.
2 Multiplicative Number Theory
We begin with a famous result due to Hardy and Ramanujan [11]
concerning
the function w, which counts the number of divisors of an integer
w (n) =  { p such that pn } .
Here and in the sequel, p always denotes a prime number.
Theorem 1
Let Y (x) ® ¥. Then the set of integers such that
 w (n)  log log n  £ Y (n) (log log n )^{1/2}
(1)
has density one.
This result means that

 { 1 £ n £ N: (1)
holds }  

N 

=1.

The original proof is quite technical. We quote here an efficient way to get
the result which is due to Turán [17].



= 
N^{1} 

(N/p +O(1))= 

1/p +O(1)= log log N +O(1) 

. 

See for example [16] for the last equality. In the same way, we get
N^{1} 

w (n)^{2}=(log log N)^{2}+ O(log log N),

thus the number of exceptions to (1) up to N is
 { 1 £ n £ N:  w (n)  log log n ³ Y (n) ( log log 
n)^{1/2} } 
£ 


(w (n)  log log n)^{2} 

Y(n)^{2} log log n 

,

which is O(N/ Y(N)^{2}). This proves the result.
Now let us show a probabilistic approach. Let (X_{p}) be a family of independent random variables
defined by
X_{p}= 
ì í î 
1, 
with probability 1/p, 
0, 
with probability 11/p. 


The mathematical expectation of w = å_{p £ N}X_{p} is
E(w)= 

E(X_{p})= 

1/p= log log N +o(log log N),

and the variance of w is
V(w)= 

V(X_{p})= 

(11/p)/p=
log log N +o(log log N).

Chebyshev's inequality yields
Pr 
{ 
 w  E(w)  ³ Y (V(w))^{1/2} 
} 
£ 1/ Y^{2},

that is the result.
This result can be interpreted, with a probabilistic point of view, as a weak law of large numbers.
A much more precise result, that is a central limit theorem, has been
proved by Erdös and Kac in 1939 [4, 5].
Theorem 2
For u a real, one has


 { 1 £ n £ N: w (n)  log log n £ u (log log 
n )^{1/2} } 
¾® 




ó õ 

e 

dt.
(2) 
A natural question is now: can this result be extended to other (strongly) additive functions (that is such that
f(n)=å_{pn}f(p))? More precisely, can one compare (and how far?)

 
{ 
1 £ n £ N : f(n) £ u 
} 

with
Pr 
ì í î 

X_{p} £ u 
ü ý þ 
,

where the X_{p}'s are independent Bernoulli random variables such that Pr { X_{p}= f(p) } =1/p and
Pr { X_{p}= 0 } =11/p.
By sophisticated sieve arguments, it can be shown that

{ 1 £ n £ N : 

f(p) £ u} 
Pr 
{ 

X_{p} £ u} =
O 
é ê ê ë 
x^{1/15}+
exp 
æ ç ç è 
 



log 


ö ÷ ÷ ø 
ù ú ú û 
,

which is o(1) when r=x^{e (x)} tends to + ¥ but e (x) ® 0. This
model is called the ``Kubilius model''.
3 Additive Number Theory
3.1 Sidon problem
In 1932, Sidon raised the following question:
is it possible to find a sequence A of nonnegative integers such that

(i) For every positive integer n, there exist a,b in A such that n=a+b;
 (ii)  {a, na Î A } = o(n^{e}) for any e >0.
In other words: is it possible to build a ``thin'' basis of order 2?
In 1954, Erdös [3] answered positively the question, by proving a more precise
result.
Theorem 3
There exists A and 0< c_{1} < c_{2} such that
c_{1} log n <  { a, na Î A }  < c_{2} log n,
(3)
for n ³ 2.
The proof is probabilistic and particularly short but absolutely not
constructive:
let (W, T,P) be a
probabilistic space and X_{2},X_{3},... be independent Bernoulli
random variables
such that Pr { X_{n}=1 }= c ( (log n) /n)^{1/2}. For w Î W, let
A(w)= { 0,1 } È {n: X_{n}(w)=1 }. Then for almost all
w Î W, A(w) satisfies
(3). A very detailed proof of this result is given in
the book by
Halberstam and Roth [9, chap. 3].
3.2 Restricted partition function
This example underlies the probabilistic interpretation of
the powerful RamanujanHardyLittlewood circle method [18].
For a given N, let q(N) be the number of ways to write
N= n_{1}+n_{2}+...+n_{r},
with 0<n_{1}<n_{2}<...<n_{r} (£ N). Another way to say this is: let
E_{N}= { 0,1 } ×...× { 0,N },
then
q(N)= 2^{N} 
 { x Î E_{N} , å x_{n}=N } 

 {x Î E_{N} }  

.

Let X_{1},...,X_{N} be independent random variables such that X_{n} takes the values 0 and n with
probability 1/2. We have
q(N)= 2^{N} Pr { X_{1}+...+X_{N}=N }.
Denote E_{N} and V_{N} the expectation and the
variance of X_{1}+...+X_{N}. If we assume that a local limit theorem holds, then we can write
q(N)= 2^{N} 


æ ç ç è 
exp 
æ ç ç è 
 


ö ÷ ÷ ø 
+o(1) 
ö ÷ ÷ ø 
.

An easy computation shows that E_{N} ~ N^{2}/4, V_{N} ~ N^{3}/12 and that the exponential
term is o(1) and so that
q(N)= o(2^{N}N^{3/2}) which is not interesting. This is due to the fact that N is too far from E_{N}.
This approach is far from being perfect. The reason is quite clear: in the problem of partitions for
an integer N, the integers 1,2,...,N do not have the same importance: the small values take part
much more frequently in a decomposition of N than the large ones. Thus we have to refine the
model by weighting the different integers, with a smaller weight for large integers.
Suppose now the X_{i}'s are independent random variables taking only the
values 0 and n with Pr { X_{n}=n } =p_{n}. Then
q(N)= 


å 
x Î E_{n}, x_{1}+...+x_{N}=N 

Pr { (X_{1},...,X_{N})=x } 




.

If we take p_{n}= exp( s n)/(1+ exp ( s n)), for some s such
that E(X_{1}+...+X_{N})=N, that is to say
we can prove (this is naturally the heart of the matter) that
q(N) ~ 

exp (p N^{1/2}/ (3)^{1/2})

(see [10, 13]). This argument has been recently developed in [8] in a more
general context.
Proof goes as usual, by defining the characteristic function (i.e., the Fourier transform of the
image measure)
f_{n}(t)=(1p_{n})+p_{n} exp(2 p i nt).
Then
Pr 
{ X_{1}+...+X_{N}=N } = 
ó õ 


æ ç ç è 

f_{n}(t) 
ö ÷ ÷ ø 
exp (2 p i Nt) dt.

The main term (corresponding to major arc) comes from a neighbourhood
of 0 on the torus R / Z. There ``remains'' (it is not easy!) to
find an upper bound for
 Õ_{n £ N} f_{n}(t) outside this neighbourhood, that is on the minor arc.
3.3 Probabilistic models for sums of powers
It is well known that sums of two squares have zero density. But for s ³ 3,
nothing is known about (lower) density of the set of sums of s integral sth
powers, like sums of 3 cubes and of 4 biquadrates.
There are two conjectures. In 1968, Barrucand [1] conjectured that for s=3 and 4 the
answer is NO. But, in 1986, Hooley [12] conjectured that the
answer is YES for every s ³ 3.
In order to guess something in this hard problem, Erdös and Rényi [6], in 1960,
considered ``pseudo sth powers'', i.e., random sequences A^{(s)} defined
as in the answer to Sidon's question, and suggested that the number of
representation r_{s}(N) of an integer N as a sum of s elements from
A^{(s)} should follow (almost surely) a Poisson law. Unfortunately, their proofs
contained a gap because of the difficulty of dealing with the quasiindependence of the sets involved.
In 1965, Halberstam and Roth [9] overcame the difficulty when s=2, by combinatorial
arguments. In 1995, Landreau [14] proved a correlation
inequality, having its own probabilistic interest, which leads to the expected result.
Theorem 4
Let E_{1},..., E_{N} be independent events, and A_{1},...,A_{T} be such that
each A_{t} is an intersection of some of the E_{n}'s. Then
0 £ Pr 
(Ç 

) Õ Pr 
( 

) £ 


( 
Pr (A_{t} Ç A_{t'}) Pr (A_{t}) Pr (A_{t'}) 
) 
.

4 Probability
We illustrate here the possibility of applying number theory ideas to probability theory with the
following recent theorem taken from [2], of which we sketch the proof.
Theorem 5
Let log 4/log 3< s £ 2, n Î N^{*} and e >0, X_{1},...,X_{n}
be i.i.d. integral valued random variables such that







Pr { X_{1}=l } £ 1  e
(4) 
and for all L ³ 2
where Q(Y,x) denotes sup_{h Î }_{R} Pr { h< Y £ h+x }. Then there exists a
constant c=c(s, e, Q(X_{1}, 1)) such that
Q(X_{1}+...+X_{n};1) £ cn 

.

The proof, in the same manner as above, uses the characteristic function f of X_{1}.
If S_{n}=X_{1}+...+X_{n}, we have
Pr 
{ S_{n} =k } = 
ó õ 

f (t)^{n} exp (2 p i kt) dt,

so that Q(S_{n},1) £ ò_{0}^{1}  f (t)^{n} dt.
We are now reduced to study the large values of f. We first use a
lemma which has been introduced in [15] (and which follows from
Bochner's theorem).
Lemma 1
Let E_{q}= { t Î R / Z,  f (t) ³ cos q }, where R / Z is once again the
torus, we have for q_{1}, q_{2} ³ 0 and q_{1}+ q_{2} £ p /2,
Then we need a result by Freiman [7] on structure of set addition (it
has been extended to the torus in [15]).
Theorem 6
Let A be a finite subset of Z such that
 A + A  £ 2  A 1 +b,
with b £  A3, then there exists an arithmetic progression L with
 A+b elements that contains A.
This enables us to show that either the set of the arguments for which f is large has a small measure,
or it has a structure (it is located close to the vertices of a regular polygon),
which is not possible in view of (4) and (5).
References
 [1]

Barrucand (Pierre). 
Sur la distribution empirique des sommes de trois cubes ou de quatre
bicarrés. ComptesRendus de l'Académie des Sciences, vol. 267,
1968, pp. 409411.
 [2]

Deshouillers (J.M.), Freiman (G. A.), and Yudin (A. A.). 
On Bounds for the Concentration Function, 1. 
Prépublication n°M/95/37, IHES, 1995.
 [3]

Erdös (P.). 
On a problem of Sidon in additive number theory. Acta
Scientiarum Mathematicarum Szegediensis, vol. 15, 1954,
pp. 255259.
 [4]

Erdös (P.) and Kac (M.). 
On the Gaussian law of errors in the theory of additive functions.
Proceedings of the National Academy of Sciences of the USA, vol. 25,
1939, pp. 206207.
 [5]

Erdös (P.) and Kac (M.). 
The Gaussian law of errors in the theory of additive number
theoretic functions. American Journal of Mathematics, vol. 62,
1940, pp. 738742.
 [6]

Erdös (P.) and Rényi (A.). 
Additive properties of random sequences of positive integers. Acta Arithmetica, vol. 6, 1960, pp. 83110.
 [7]

Freiman (G. A.). 
Foundations of a structural theory of set addition. 
American Mathematical Society, Providence, R. I.,
1973, Translations of Mathematical Monographs, vol. 37.
Translated from the Russian.
 [8]

Freiman (Gregory A.) and Pitman (Jane). 
Partitions into distinct large parts. Journal of the Australian
Mathematical Society, vol. 57, n°3, 1994, pp. 386416.
 [9]

Halberstam (Heini) and Roth (Klaus Friedreich). 
Sequences. 
SpringerVerlag, New YorkBerlin, 1983, 2nd edition.
 [10]

Hardy (G. H.) and Ramanujan (S.). 
Asymptotic formulæ in combinatory analysis. Proceedings of
the London Mathematical Society, vol. 16, 1917, pp. 75115.
 [11]

Hardy (G. H.) and Ramanujan (S.). 
The normal number of prime factors of a number n. Quarterly
Journal of Mathematics, vol. 48, 1917, pp. 7692.
 [12]

Hooley (Christopher). 
On some topics connected with Waring's problem. Journal
für die Reine und Angewandte Mathematik, vol. 369, 1986,
pp. 110153.
 [13]

Hua (LooKeng). 
On the number of partitions of a number into unequal parts. Transactions of the American Mathematical Society, vol. 51,
1942, pp. 194201.
 [14]

Landreau (Bernard). 
Étude probabiliste des sommes de s puissances sièmes. Compositio Mathematica, vol. 99, n°1, 1995, pp. 131.
 [15]

Moskvin (D. A.), Freiman (G. A.), and Judin (A. A.). 
Inverse problems of additive number theory and local limit theorem
for lattice random variables. In Numbertheoretic studies in the
Markov spectrum and in the structural theory of set addition,
pp. 148162. 
Kalinin. Gos. Univ., Moscow, 1973.
 [16]

Tenenbaum (Gérald). 
Introduction to analytic and probabilistic number theory. 
Cambridge University Press, Cambridge, 1995, Cambridge Studies in Advanced Mathematics, vol. 46. Translated from the
second French edition (1995).
 [17]

Turán (P.). 
On a theorem of Hardy and Ramanujan. Journal of the London
Mathematical Society, vol. 9, 1934, pp. 274276.
 [18]

Vaughan (R. C.). 
The HardyLittlewood method. 
Cambridge University Press, Cambridge, 1997, 2nd
edition, Cambridge Tracts in Mathematics, vol. 125.
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