Integers without large prime factors

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

A

DOLF

H

ILDEBRAND

G

ERALD

T

ENENBAUM

Integers without large prime factors

Journal de Théorie des Nombres de Bordeaux, tome

5, n

o

_{2 (1993),}

p. 411-484

<http://www.numdam.org/item?id=JTNB_1993__5_2_411_0>

© Université Bordeaux 1, 1993, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Integers

without

_large

_prime

factors

by

ADOLF HILDEBRAND* AND

GERALD

TENENBAUM

0. Introduction

1. Estimates a _survey

2. The Dickman function

3.

_y)

for small u

4.

_~(x,

_y)

for

_large

u

5. Distribution in short intervals

6. Distribution in arithmetic

_progressions

7. Other results

o. Introduction

The _purpose of this _{paper is} to

_give

a _survey of recent work on the

distri-bution of

_integers

without

_large

_prime

factors. A similar _survey had been

published

about

_twenty

_{years ago}

_by

K. Norton

_(1971),

but in the

interven-ing

time the

_subject

has been

_considerably

advanced and is now in a mature

and

_largely

_satisfactory

state.

_Moreover,

the results have found some new

and rather

_{unexpected applications}

in diverse areas of number

theory.

We

therefore felt it

_appropriate

to

_give

an account of the

_present

state of the

subject

which _may be useful for those interested in

_studying

the

_subject

for its own sake as well as those interested in

applying

the results. While

integers

without small

_prime

factors _may be viewed as

_{approximations}

to

primes

and as such form a natural

object

of

study

in

prime

number

theory,

the reasons for

studying integers

without

large

prime

factors are less

ob-vious.

_Integers

without

_large

_prime

_factors,

also called "smooth"

_integers,

are

_{necessarily products}

of _many small

_prime

factors and _are, in a _sense,

the exact

_opposite

of

_prime

numbers.

_However,

it turns out that results

on

_integers

without

_large

_prime

factors

_play

an

_important

auxiliary

role

in several

_problems

in

_prime

number

_theory,

in

_particular

in the

construc-tion of

_large

_{gaps between}

_primes

_{(Rankin (1938))}

and in the

_analysis

of

algorithms

for

_factoring

and

_{primality testing}

_{(Pomerance (1987),}

Lenstra

(1987)).

They

are also relevant in some other

problems

in number

theory

Manuscrit regu le 15 mars _1993, version definitive le 28 d6cembre 1993.

*Supported in _{part by} an NSF Grant while _visiting the Institute for Advanced _Study at

such as bounds on the least kth _power non-residues

(Vinogradov (1926),

Norton

_{(1969,1971)),}

_{Waring’s problem}

_{(Vaughan (1989),}

_Wooley

_(1992)),

and Fermat’s

_conjecture

_{(Lehmer &#x26;}

Lehmer

_(1941),

Granville

_(1991b)).

Most

_recently,

a result on

integers

of the form _p + 1 without

_large

_prime

factors

_played

an essential role in the resolution of a

long-standing

conjec-ture of Carmichael

_(Alford,

_Granville,

&#x26; Pomerance

_(1993)).

In

_addition,

the

_study

of

_integers

without

_large

_prime

factors is an

interesting

and

diffi-cult

_problem

for its own sake that can be attacked

_by

a

_variety

of

_methods,

some of which have led to advances on other

problems

in

analytic

number

theory.

The

_{principal object}

of

_{investigation}

is the function

where

_P(n)

denotes the

_largest

_prime

factor of _n, with the convention that

P(l)

= 1. The ratio

iY(x,

y)j[x]

_may be

interpreted

_as the

probability

that a

_randomly

chosen

_integer

from the interval

_{~1, ~~}

has all its

_prime

factors _y.

Non-trivial estimates for

_{q, (x, y)}

can be obtained

by

a

variety

of

methods,

depending

on the relative size of _y and x and the nature of the desired result. In Section 1 we shall _survey the

_principal

results and discuss some of the

methods of

_proof.

Section 2 will be devoted to the Dickman

_function,

a

function defined

_by

a differential-difference

_equation

which arises in this

connection. In Sections 3 and 4 we shall

give complete proofs

of two of the main results on In Section 5 we consider the distribution of

integers

without

_large

_prime

factors in short intervals and _prove a new result

in this context

_(Theorem

_5.7)

which extends a recent result of Friedlander &#x26;

Lagarias.

Section 6 is devoted to the distribution in arithmetic

_{progressions,}

and in the final section we _{survey various} other results on

_integers

without

large

prime

factors. We conclude with a

comprehensive bibliography

of

papers on the

subject

that have

_appeared

since the

publication

of Norton’s

memoir. The reader _may find another

_quite

_thorough

list of references in

Moree’s thesis

_(1993),

which also incudes a clear introduction to the matter

as well as

interesting

new contributions.

Notation. We let

_{logk x}

denote the kfold iterated

_logarithm,

defined

by

log, x

=

log

_x and

logk x

=

loglogk_1 x

for k _&#x3E; 1. Given _a

complex

num-ber _s, we denote its real and

_imaginary

_parts

_by

a and _T,

respectively.

The letter E denotes as usual an

arbitrarily small,

but fixed

_constant;

other

constants will be denoted

_by

_{c, yo,}

_etc.,

and need not be the same at each

occurrence. We use the notations

_f(x)

=

0(g(x))

and

f(x) «

x in a _range which will

_normally

be clear from the context. The constant c

here is allowed to

_depend

on c

(if

the functions involved

depend

on

E),

but

any

dependence

on other

_parameters

will be

_explicitly

indicated

_{by writing}

Ç)k,

~71’

etc. We write

_{I(x) x g(x)}

if both

_{f (x) «}

_g(x)

and

_{f (x)}

hold.

Acknowledgements.

We are

grateful

to Jean-Marie

_DeKoninck,

Heini

Halberstam,

Aleksandar

_{Ivic, ,}

Pieter

_Moree,

Karl

_Norton,

Carl

_Pomerance,

Eric

_Saias,

J.

_Sandor,

and Jie Wu for their comments on earlier versions of

this _paper.

1. Estimates for

_{~(:r,~) 2013}

a _survey

We

_begin

_{by pointing}

out a natural but false heuristic for the size of

BII ( x, y).

It consists of the

_{approximation}

which is

_suggested

_by

a

simple probabilistic

_argument

and also

by

"extrap-olation" from the sieve estimate

which holds for _any set of

_{primes P}

contained in

_{[1, xl/2-E]}

_(see,

_e.g.,

Hal-berstam &#x26; Ri chert

_(1974))

_By

Mertens’

_theorem,

the

_right-hand

side of

(1.1)

is of order

_x/u

with u =

log x/ log y.

However,

as the results below

show,

T(x, y)lx

is in fact

_{exponentially}

_decreasing

in _u, and therefore of much smaller order of

_magnitude.

The reason for this

discrepancy

is that

the

_validity

of

_(1.2)

_depends

on certain

independence

assumptions

which

are not satisfied if the set P contains

_large

_primes.

For

_example,

_primes

in

the interval

_x~

do not act

_{independently}

in the sense that if an

_integer

n x is divisible

_by

one such

_prime

then it cannot be divisible

_by

_any other

prime

in this interval. We remark that in many

applications

it is

_precisely

this

_discrepancy

between the

_expected

and the actual size of

_{BIf(x, y)}

which

is

_exploited.

The failure of the heuristic

_(1.1)

shows that classical sieve methods are not an

_appropriate

tool for

_estimating

_{T(x, y).}

_Indeed,

these methods

would lead to an

_{approximation}

for

_consisting

of a main term of the order of the

_right-hand

side in

_(1.1),

and an error term which in view of the above remarks would have to be of at least the same order of

magnitude

as the main term. Thus one cannot

_hope

to obtain a lower bound for

T(x,

y)

in this _way. An _upper bound

_by

the

_right-hand

side of

_(1.1)

can be

_deduced,

We now derive a

_simple,

but nonetheless useful bound for

y),

using

a

technique

known as "Rankin’s

method";

see Rankin

(1938).

It is based on the observation

that,

for _any &#x3E;

_{0, x &#x3E; 1,}

and y &#x3E;

_2,

If we take u = 1-

_{1/(2log y),}

then the last sum _may be estimated

by

Substituting

this into

_(1.3)

_gives

the bound

where here and in the

_sequel

we set u =

log x/ log y.

By using

_a

slightly

more

complicated

_argument

one can remove the factor

logy

in

(1.4);

see

Theorem 111.5.1 in Tenenbaum

_(1990a).

An

_asymptotic

formula for

_{Bl1(x, y)}

was first obtained

by

Dickman

(1930)

who

_proved

that for _any fixed u &#x3E; 0

where the function is defined as the

(unique)

continuous solution to

the differential-difference

_equation

satisfying

the initial condition

(Actually

Dickman stated his results in terms of the function

#{n

x :

_P(n)

_{which, however,}

is

_easily

seen to be

asymptotically

equal

to

_{W(z, y) when u}

is

_fixed.)

A more

_{rigorous proof,}

by

modern

stan-dards,

was later

_{supplied by}

Chowla and

_{Vijayaraghavan}

_(1947);

the first

quantitative

estimates were obtained

_by

Buchstab

(1949)

and Ramaswami

(1949).

The "Dickman function"

_p(u)

is

_nonnegative

for u &#x3E;

_0,

_decreasing

for

u &#x3E;

_1,

and satisfies the

_asymptotic

estimate

These and other

_properties

of the Dickman function will be

_proved

in

Sec-tion 2.

Substantial _progress on the

_problem

of

_estimating

_y)

was made in

the 1950s

by

de

_Bruijn.

_Among

other

_estimates,

de

_Bruijn

_(1951b)

_proved

a uniform version of Dickman’s

result;

when combined with the

sharpest

known form of the

_prime

number

_theorem,

his result states that

holds

_{(with u}

=

_log

x /log y)

uniformly

in the

range

While the error term in

(1.8)

is

best-possible,

Hensley

(1985)

showed that a lower bound of the form

W(z, y)

&#x3E;

_xp(u)

holds in a much

larger

_range,

and the range of

validity

for

_(1.8)

itself has been

_{significantly improved by}

Maier

_{(unpublished)}

and Hildebrand

_(1986a).

The latter result

_gives

the

largest

range in which the

asymptotic

relation

xp(u)

is known to

hold,

and we state it

_formally

as a theorem.

THEOREM 1.1. For _any

_fixed

c &#x3E; 0 the relation

_(1.8)

holds

_uniformly

in

the _range

The _upper limit in _{this range is}

_closely

tied to the best known error

term in the

_prime

number

_theorem,

and _any

_improvements

in the error

term would lead to

_{corresponding}

_improvements

in the _range

_(1.10).

In

fact,

the

_{correspondence}

is in both

_directions;

for

_example,

one can show

that

_(1.8)

in the form =

xp(u) exp

10(log(u

+

_1)/logy)}

holds

uniformly

in the _range

_{y &#x3E;}

_2,

1 u

_{yl/2-f., for}

_any fixed c &#x3E;

_0,

if and

only

if the Riemann

_Hypothesis

is

true;

see Hildebrand

(1984a).

The

proof

which holds for

_{any x &#x3E;}

1

_{and y &#x3E;}

2. This

_equation

is obtained

_by

evaluating

the sum S = in two different _ways. On the

one

hand, partial

summation shows that S is

equal

to the difference between the left-hand side. and the first term on the

_right

of

_(1.11).

On the other

hand, writing log n

log p

and

_inverting

the order of

_summation,

we see that S is also

_equal

to the sum on the

_right

of

_(1.11).

The estimate

(1.8)

is obtained

_by

first

_showing

that it holds in an initial range,

_{say x}

y2,

and then

_applying

_(1.11)

_iteratively

to show that it continues to hold in the range x

y(~+1)/2

for n &#x3E; 4 as

_long

as

_(1.10)

is satisfied. The

limitation in the _range comes from the fact that each iteration

_step

involves

a small error due to

_{possible irregularities}

in the distribution of

_primes.

These errors accumulate and after

sufficiently

_many iterations become as

large

as the main term.

The

_principal

difference between this

_approach

and the earlier

_approach

of de

_Bruijn

lies in the use of

_(1.11).

De

_Bruijn

based his

_argument

on a

different functional

_equation,

the "Buchstab

_identity"

which holds for all x &#x3E; 1

_{and z &#x3E; y}

&#x3E; 0.

_Compared

to

_(1.11),

this

_equation

has the

_disadvantage

that the second

_argument

of T is not

_fixed,

so that an

induction

_argument

based on this formula is

technically

more

complicated.

A more severe limitation of the Buchstab

identity

is that the

right-hand

side consists of two terms

_having

_opposite

_signs

but which can be

_nearly

equal

in size. As a

_result,

the error terms

_arising

from inductive use of

the Buchstab

_identity

are much

_larger

than those

_coming

from

_(1.11),

and

exceed the main term

_already

for

_relatively

small values of u.

Using

the functional

_equation

_(1.11),

it is

_relatively

easy to

_justify,

at

least

_{heuristically,}

_{the appearance of} the Dickman function in

_(1.8)

if we

suppose that a relation of the form

_{~Y(x, y) r·· x f (u)}

holds with some

func-tion

_{f (u).}

_{Replacing w(t,y)}

_by

_{t f (log t/ log y)}

in

_(1.11),

we obtain under

this

_assumption

_{y &#x3E;}

2

Since x, we

necessarily

have

f (u)

_1,

so that the first term

on the

right-hand

side is of order

_O(x).

_Moreover,

the contribution of the

order of

_magnitude.

After

_dividing

_by

_{x log x,}

both of these contributions become

_o(l)

and we obtain

using

the

_prime

number theorem in the form

_{?P(t) -}

t.

_Assuming

the

error terms involved all tend to zero as _{x, y -~ oo,} we conclude that the

function

_{f (u)}

satisfies the

_integral

_equation

_{f (u)u}

_{f (v)dv}

for u &#x3E; 1.

Differentiating

both sides of this

_equation

shows that

_{f (u)}

is a solution

to the differential-difference

_equation

_(1.5).

Since for 0 u 1 we have

trivially

~(yu, y) _

as y --&#x3E; oo,

f(u)

also satisfies the initial

con-dition

_(1.6).

_Therefore,

_f(u)

must be

_equal

to the Dickman function

_p(u).

The _range

_(1.10)

is the

_largest

in which an

_asymptotic

_{approximation}

for

’I!(x, y)/x

by

a smooth function is known. This _range can be

considerably

increased if instead of an

asymptotic

formula for we

only

ask for

an

_asymptotic

formula for as the

following

result shows.

THEOREM 1.2. For _any

_fixed

c &#x3E; 0 we have

uniformly

in the _range

’

Moreover,

the lower bound in

_(1.12)

is valid

_{uniformly for}

all x &#x3E;

_{y &#x3E;}

2.

The _upper bound in this result is

_implicit

in de

_Bruijn

_(1966),

whereas the lower bound is due to Hildebrand

_(1986a).

A

_slightly

weaker result had been

_{given by Canfield,}

Erd6s &#x26; Pomerance

_(1983).

Combining

(1.12)

with the

_asymptotic

formula

_(1.7)

for the

_logarithm

of the Dickman

_function,

we obtain the

following

simple,

but

_quite

useful

corollary.

COROLLARY 1.3. For any

fixed

c &#x3E; 0 we have

In terms of the variables x and _y, the _ranges

_(1.10)

and

_(1.13)

of

Theo-rems 1.1 and 1.2

correspond

to

respectively.

To

_give

an idea of the size of

y)

in various

_parts

of these

ranges, we consider the cases _y = _exp with 0 a 1 and

y =

(log

x)"

with _{a &#x3E;} 1. The

corollary

yields

in the first _case

and in the second case

The estimate

_(1.14)

_together

with the

_monotonicity

of shows that

_{I log}

_{~~(x, y)jx)}

_I

is of order

_log

x whenever _y is smaller than a fixed

power of

log x.

In this range it is therefore more

appropriate

to seek an

approximation

to

_log

_{~~(x, y)~}

rather than to Such an

estimate has been

_{given by}

de

_Bruijn

_(1966);

his

_result,

in a

slightly

more

precise

form due to Tenenbaum

_(1990a,

Theorem

_111.5.2),

is as follows.

THEOREM 1.4.

_{Uniforrrtly for}

x &#x3E; _y &#x3E;

_2,

we have

where

The

_proof

of this result is

_{completely elementary.}

The _upper bound is deduced from Rankin’s

_inequality

_(1.3)

with an

optimal

choice of the

parameter u. The lower bound is based on the

_elementary

_inequality

together

with a lower bound for the

right-hand

side of

_(1.17)

obtained

by

Stirling’s

formula. To _prove

_(1.17),

it suffices to note that the

_right-hand

side

_represents

the number of

_k-tuples

_{m2, ... ,}

_mk)

with 0 and

Mi = l and

that,

by

the definition of k and the Fundamental Theorem

of

_Arithmetic,

the set of such

_tuples

is in one-to-one

_{correspondence}

with the set of

_{integers composed}

of

_{exactly i}

_(not

_necessarily

_distinct)

_prime

factors

_y),

and hence with a subset of the set of

integers

counted

in

_{T(x, y).}

,

The estimate

_(1.15)

_clearly

shows that there is a

_change

of behavior for

W ( x, y)

at y s5

log x.

If _oo, then the first term in the definition of Z

_dominates,

whereas for _{y =}

_{o(log x),}

Z is

_asymptotic

to the second

term in

_(1.16).

The

_change

in behavior is due to the fact that if _y is small

compared

to

_{log x,}

then _many

_prime

factors of a

"typical" integer

counted

occur to

_high

_powers, whereas for

_larger

values

_{of y}

most

_prime

factors occur

only

to the first _power. For a closer

analysis

of T(x, y)

near

the transition

_point

_{y s5}

_log

x see Granville

_(1989).

In the case of _very small values of _y, it is useful to observe

_that,

_by

the Fundamental Theorem of

_Arithmetic,

is

_equal

to the number of

solutions

_(mp)p~y

in

_nonnegative

integ~rs

to the

_inequality

or

equivalently

the linear

diophantine inequality

The number of solutions to this

_inequality

can be estimated _very

precisely

by elementary geometric

methods as

long

as the number of variables

(i.e.,

~r(y))

is not too

_large.

For

_example,

it is easy to see that

if y &#x3E;

2 is fixed

and x ~ _oo, then the number of solutions is

_{asymptotically equal}

to the volume of the

_{x(y)-dimensional}

_simplex

defined

_by

the

_inequalities

where k =

7r(y)

and

pi denotes the ith

prime. By

a

change

of

variables,

the

volume of

_(1.19)

is seen to be

_equal

to

I- L

I-General

_asymptotic

results of this

_type

have been

_{given by Specht}

_(1949),

Hornfeck

_(1959),

Beukers

_(1975),

Tenenbaum

_(1990a)

-

Theorem 111.5.3

-

and Granville

_(1991a).

A more careful

_reasoning

leads to the

_following

THEOREM 1.5.

Uniformly

in the _range

we have

This result is stated in Ennola

_(1969),

who has also

_given

a

similar,

but

somewhat more

complicated

formula for the

larger

_{range y}

(log

x)3~4-E;

for a detailed

proof

see Theorem 111.5.2 of Tenenbaum

(1990a).

Theorems 1.1 and 1.5 both

_give

an

_asymptotic

formula for

’li(x, y);

the

first result is valid for

_relatively

_large

values

_{of y}

and

_gives

an

approximation

by

a smooth

function,

whereas the second result holds for

small y

and

gives

an

_{approximation by}

a

_quantity

depending

on the

primes

_y. Between

the two _ranges

_(1.10)

and

_(1.20),

_however,

there remains a

large

_{gap in}

which the results

_quoted

above

_{give only}

_{much weaker estimates. This gap} has been closed

_by

the

_following

result of Hildebrand &#x26; Tenenbaum

_(1986),

which

_gives

an

_asymptotic

formula for

_{’li(x, y)}

that is valid

uniformly

in x

and _y,

_{provided only}

that u =

log x/ log y

and _y tend _to

infinity.

THEOREM 1.6.

Uniformly in

the _{range x &#x3E;}

_{y &#x3E;}

_{2, we}

have

and a =

a(x,

y)

is the

_unique

_positive

solution to the

_equation

The formula

_(1.22)

is similar in nature to the

_explicit

formula in

_prime

number

_theory.

It _expresses in terms of the

_generating

Dirichlet series

_{C(s, y)}

evaluated at a certain

_point

a which

plays

a role

analogous

to

that of the zeros in the

_explicit

formula. Note that the function

_{(0’, y)xa}

is

_{exactly equal}

to the _upper bound

_(1.3)

for

_y)

obtained

_by

Rankin’s

method,

and that

_{a(x, y)}

is the

_{unique point}

on the

_positive

real line which

minimizes this function. Thus the denominator in

_(1.22)

measures the ratio

between

_iY(x,

_y)

and the _upper bound

_{given by}

Rankin’s method with an

optimal

choice of _parameters.

A formula of this

_type

_may seem to be of little value at first

_sight,

since

it involves the _parameter a which is defined

only implicitly

through

an

equation

involving

prime

numbers.

_Nonetheless,

from a

sufficiently sharp

form of the

_prime

number theorem one can derive the estimate

uniformly

in x &#x3E; y &#x3E;

2

_(see

Hildebrand &#x26; Tenenbaum

_(1986)),

and

_using

this

_estimate,

one can show that the

right-hand

of

(1.22)

is

approximated

by

the

_right-hand

sides of the formulas of Theorems 1.1 - 1.5 in their

respective

ranges,

except

in the case of Theorem 1.1 when u

log y,

_say.

Thus,

Theorem 1.6

_implies

Theorems 1.2 - 1.5 in their full

_strength

_(except

for the

_quality

of the error term of

_(1.21)

when

_{y W}

_x)2/3),

as well as the statement of Theorem 1.1 for u &#x3E;

_{log y.}

We

_emphasize

that Theorem 1.6 does not lead _{to any}

_improvements

in the ranges of

validity

of the above theorems. For

_example,

it does not

_yield

an

_{asymptotic approximation}

to

_by

a smooth function in the range

u &#x3E;

_exp((log

_y)3/5).

This is due to our limited

knowledge

of the distribution

of

_primes

which

_prevents

us from

obtaining

a smooth

approximation

to the

right-hand

side of

_(1.22)

in _{this range.}

_However,

even in the _range where no

smooth

_{approximation}

to

_{BII(x, y)}

is

_available,

one can use

(1.22)

to obtain

very

precise

information on the local behavior of

y).

The

following

is a

typical

result of this

_type,

which we

_quote

form Hildebrand &#x26; Tenenbaum

(1986).

COROLLARY 1.7.

_{Uniformly for x &#x3E;}

_{y &#x3E;}

2 and 1 ~

_{c y,}

we have

where

_{a(x, y)}

is

_defined

as in Theorem 1.6.

Since

_by

_{(1.26), a(x, y)}

=

0(1)

if and

only

if _y

-5

(logx)l+o(l),

this

shows,

for

_example,

that

_{iY(x, y)}

as _{x, y -+} oo holds if and

only

if’

In Section

_4,

we shall

_give

a

complete proof

of Theorem 1.6 for the _range

u &#x3E;

_(log

_y)4.

The method of

_proof

is

_analytic;

it

_depends

on

representing

T(x, y)

as a

complex

_integral

over the function

C(s,

and

evaluating

this

_integral

_by

the saddle

_point

method. The

_point

_{a(x, y)}

is a saddle

_point

for

_{( s, y)x-1,}

and the

_expression

on the

right-hand

side of

(1.22)

arises as

The

_{principal deficiency}

of Theorem 1.6 is that the error term in

_(1.22)

increases as u decreases and for small u becomes

larger

than the error term

of Theorem 1.1. Saias

₍₁₉₈₉₎

has shown that one can

adapt

the saddle

point

method to deal more

effectively

with this range. He obtained the

following

result which is valid in the same _range as Theorem

_1.1,

but

_gives

a much more

precise

estimate for

y).

THEOREM 1.8. For any

fixed

c &#x3E;

_0,

and

_{uniformly for}

_{y &#x3E;}

_yo(c)

and

1 u we have

and

_{p(u) is}

_{defined by}

_(1.5)

and

_(1.6)

_{for u &#x3E;}

0 and

_p(u)

= 0

for u

0.

We shall _prove this result in Section 3. The function

_{A(x, y)}

has been

in-troduced

_by

de

_Bruijn

_(1951b),

who obtained an estimate similar to

_(1.28),

but

_only

for a much smaller _range. In Lemma 3.1 we shall show

that,

uniformly

in y &#x3E;

yo(E)

and

_{1 u}

_y1-f.,

Thus the estimate of Theorem 1.8

_implies

that of Theorem 1.1

_{if y}

is

suf-ficiently large.

In

_fact,

one can

sharpen

(1.30),

by giving

an

asymptotic

expansion

of the O-term in _powers of

_{1/ log y,}

_resulting

in a

_{corresponding}

sharpening

of the estimate of Theorem

_1.1;

_see,

_for.

_example,

Saias

_(1989).

2. The Dickman function

In this section we

investigate

the behavior of the Dickman function

p(u)

defined

_by

_(1.5)

and

_(1.6).

Our

_principal

result is the

_following

_asymptotic

estimate for

_p(u).

THEOREM 2.1. For u &#x3E; 1 we have

where _y is Euler’s

_{constant, ~ _}

_~(u)

is the

_unique

_positive

solution to the

equation

The

_asymptotic

formula

_implicit

in

_(2.1)

has been first

_{proved by}

de

Bruijn

(1951a)

using

contour

_integration

and the saddle

_point

method. Canfield

₍₁₉₈₂₎

_gave a combinatorial

_proof

of this

formula,

and an

arith-metic

_proof

via the function

_{iY(x, y)}

is contained in Hildebrand &#x26; Tenen-baum

_(1986).

The above

_quantitative

result is due to Alladi

_(1982b)

who used de

_Bruijn’s

method. It has been

_{sharpened by}

Smida

_(1993),

who

essentially replaces

the factor 1 +

_0(1/u)

in

_(2.1)

_by

a function

_having

an

explicit

asymptotic expansion

in terms of

_(negative)

_powers of u and _powers of

_~(u).

Another

_type

of

_expansion,

as a

_convergent

series of

analytic

func-tions,

may be derived from a

general

theorem of Hildebrand &#x26; Tenenbaum

(1993)

on the solutions of differential-difference

_equations.

A

_different,

but

quite

complicated

expansion

has

_recently

been

_{given by}

Xuan

_(1993).

The

proof

we shall

_give

for Theorem 2.1 here is taken from Tenenbaum

(1990a).

We remark that

_(2.1)

Call be written as

a relation which is often more convenient to work with. This follows from the

_identity

which can be seen on

_noting

that

by

(2.2),

and

since

_{~(~) -~}

0 as u - _1+.

The function

_~(u)

is

_{non-elementary,}

but it is not hard to obtain an

expansion

for this function in _powers of

_{log u}

and

₁₀₉₂

u. We shall _prove a

simple

result of this kind.

Proof.

Let

_{f (x) _ (ex - 1)/x}

= fo

The function

_{f (x)}

tends to 1 as x --3

0+,

and it is

strictly increasing

in x since

This shows that for each u &#x3E; 1 the

_equation

_{f (~) _ ~c}

_(i.e.,

_(2.2))

has a

unique

solution ~ =

~(u)

&#x3E; 0. The

_monotonicity

of

_{f together}

with the observation that we have for _any

sufficiently large

constant c and &#x3E;

_1,

writing V

=

log2(u

₊

2)/log(u

₊

2),

implies

the estimate

_(2.4).

Differentiating

both sides of

_(2.2)

with

_respect

to u

_gives

and hence the first

_equality

in

_(2.5).

The second

_equality

follows from the first and the estimate

_(2.4),

_{provided u}

is

_{sufficiently large.}

To

_complete

the

_proof

of

_(2.5),

it suffices to show that

_1/u

holds

_uniformly

in

u &#x3E; 1. We have

and since

_~’(u)

=

1//~(~),

we obtain

_1/u

~’(u)

_4/u

as

required.

Substituting

the estimates of the lemma into

_(2.1)’

we obtain the

follow-ing corollary.

COROLLARY 2.3. F’or u &#x3E; 1 we have

The estimates

_(2.4)

and

_(2.6)

can be refined and one can in

principle

approximate

_~(u)

and for _any

_given

k &#x3E; 1

_by

an

elementary

function to within an error of order

0((IogU)-k);

_see, for

_example,

de

Bruijn

_(1951a).

Many applications require

estimates for ratios of the form

_{p(u -}

_v)/p(u)

rather than estimates for

_p(u)

itself. Theorem 2.1 leads to the

_following

result of this kind.

COROLLARY 2.4. For u &#x3E; 2

_{and Ivl}

_u/2

we have

Moreover,

for u

&#x3E; 1 and 0 v u we have

To prove this

result,

we

apply

(2.1)’

with u and with u - _v,

_getting

for 0 v u - 1.

_If,

in

_addition,

0 v

_u/2

then 1 u - v x u and hence

Differentiating

the relation

_{f (~)}

= u twice with

_{respect to u,}

we obtain

By

Lemma 2.2 and the

_inequality

_If"(ç)1

=

f(£)

_{= u} this

implies

g"(u) «

1/u2.

It follows

_that,

for u &#x3E; 2 and 0

_v

_~/2,

... , , , -

, , ,

, ,

Together

with

_(2.10)

and

_(2.9)

this

_implies

_(2.7)

in the case 0 v

_u/2,

and a similar

_argument

_gives

_(2.7)

when

_{-u/2 v}

0.

To _prove

_(2.8),

we _may assume that 0 v u -

_1,

since the

_right-hand

side is an

_increasing

function of _v, while the left-hand side is

equal

to 1 for

u - 1 v u. We can therefore

_a,pply

the first estimate of

_(2.10).

_By

Lemma 2.2 the

_{integral involving}

_~’(t)

is of order W

_{J:-v(t - u}

+

_v)/tdt

»

v2/u,

whereas,

for 0 v u -

_1,

Thus,

for a suitable constant c &#x3E;

_0,

the

_right-hand

side of

_(2.9)

is bounded

by

and

_(2.8)

follows.

We now

proceed

_{to prove} Theorem 2.1. We

begin by establishing

some

elementary properties

of the function

_p(u).

It is convenient here to set

p(u)

= 0 for u

_0,

_so that

LEMMA 2.5. We have

Proof.

The first relation holds for 0 u 1 since

_by

definition

_p(v)

= 0

for v 0 and

_p(v)

= 1 for 0 v 1. It remains valid for u _&#x3E; 1 since

p(u)

is a continuous function and

_by

(1.5)

the derivatives of both sides of

(2.11)

are

_equal

in this _range.

Inequality

(2.12)

is a _consequence of

(2.11),

(1.5),

and the

continuity

of

p(u).

For _suppose

_(2.12)

is

_false,

and let uo =

inf~u &#x3E;

0 :

p(u) 0},

so

that 1 _uo oo .

By

the

continuity

of

p(u)

this would

imply

p(uo)

= 0

and

_{u 0}

_p(u)dv

&#x3E;

_0,

which contradicts

_(2.11).

Inequality

(2.13)

follows from

_(2.12)

and

_(1.5).

_I The last

_inequality

of the lemma is true for 0 u

_1,

since

_p(u)

= 1

and

_r(u

+

₁₎

1 in this _range.

_Assuming

that it holds for k u k + 1

for some k &#x3E;

_0,

we deduce

by

(2.11)

and the

monotonicity

of

p(u)

that for

k + 1 u k + 2,

p(u)

p(u - 1)/u

1/ur(u)

= +

1).

Hence,

by induction,

(2.14)

holds for all u &#x3E;

_0,

and the

_proof

of the lemma is

complete.

We next

_investigate

the

_Laplace

transform of

_p(u),

defined

_by

By

_(2.14),

the

_integral

in

_(2.15)

is

_uniformly

_convergent

in _any

_compact

region

in the

_complex

_s-plane.

LEMMA 2.6. We have

Proof.

Using

(2.11)

we obtain

The solution to tills differential

_equation

is of the form

where C is a constant. To determine the value of

_C,

we examine the

behavior of

_p(s)

and

eE( -’)

as s ~ oo

along

the

_positive

real axis. On the

one

_hand,

the definition of

p(s)

_gives

_limu-o+

_{p(u) =}

1.

On the other

_hand,

the relation

where

(cf.

Abramowitz &#x26;

_Stegun

_(1964),

_{p. 228),}

_implies

lims-+-oo eE(-s)s

= e--t.

Hence C =

_el,

and

(2.16)

follows.

LEMMA 2.7. Let u &#x3E;

_{1, ~ =}

_~(u),

and s =

-~

_{+ ir. Then} we have

and

Proof.

The second estimate follows

_immediately

from Lemma

_2.6,

_(2.17)

The first estimate is

_by

Lemma 2.6

_equivalent

to the

_inequality

where

A

_change

of variables

_gives

For

_irl

7r and 0 t 1 we have 1 -

cos(tT)

&#x3E;

2(tr)2/7r2

and thus

The

_integral

here is

by

(2.2).

This

_gives

the first

_inequality

in

_(2.20).

To _prove the second

_inequality

in

_(2.20),

we use the estimate

For bounded values

_{of u,}

_(2.20)

holds

_trivially

since

_{H(r) &#x3E;}

0. On the other

_hand,

if _u, and

_{therefore ~,}

are

sufficiently large

and _x, then

the main term in the last

_expression

is

_greater

than or

equal

to

Proof of

Theorem 2.1. Since

_p(u)

is

_continuously

differentiable for u &#x3E; 1

and

_by

Lemma 2.5 the

_Laplace

transform

_p(s)

is

_absolutely

_convergent

for

any s, the

Laplace

inversion formula is

applicable

and

gives

for u &#x3E; 1 and _any E I(8. We choose r, =

-

with =

(u)

defined

by

(2.2).

This choice is

_{suggested by}

the saddle

_{point method,}

since the

_{point 8}

=

-g

is a saddle

_point

of the

_integrand

in

_(2.21),

_i.e.,

a zero of the function

We

_{begin by showing}

that the main contribution to the

_integral

in

_(2.21)

comes from the

range Irl

_b,

where 6 =

7rJ2 log(u

₊

I) /u.

To this end

we divide the

_remaining

_{range into} the three

_{parts b}

_7r, 7r

_Irl

ulog(u

+

_1),

and

_{Irl I}

&#x3E;

_ulog(u

+

_1).

_Using

the bounds for

_pes)

_provided

_by

Lemma

_2.7,

we obtain for the contributions of these ranges the estimates

and

respectively.

Since the

_right-hand

side of

_(2.1)

is of order

_by

Lemma 2.2 and since

_{E(~) x}

_{(et -}

_1)/~

= _u, each of these contributions

is

_by

a factor G

_1/u

smaller than the

_right-hand

side of

_(2.1)

and thus is absorbed

by

the error term in

_(2.1).

It remains to deal with the _range

_I-rl

b.

_Using

Lemma

_2.6,

we write

the

_integrand

in

_(2.21)

as

and

_expand

the function in the

_exponent

in a

_Taylor

series about T = 0.

Since

_{E’(~) _}

_{(e£ -}

_1)/~

=

From the

_inequality

we see that the two last terms in

_(2.23)

are of order

o(lrl3 u)

and

respectively,

and hence of order

₀₍₁₎

for

_{17-1 :5 6}

=

r V2 log(u

+

_1)/u.

Applying

the estimates

to the

_exponentials

of these

_terms,

we can then write

(2.22)

as

Integrating

this function over the interval -b T

b,

we obtain

since the contribution from the term to the

_integral

is zero

for

_symmetry

reasons. The contribution from the error term in

_(2.24)

is

bounded

_by

and

_extending

the _range of

_integration

in the main term to

_(-00,00)

intro-duces an error of order

and

_{2~2E"(~) _ ~r2log(u +}

_{I)) ),}

the coefficient of

6"~’~~/~/jE~(~)

in both of these error terms is of order

_1ju.

_Hence,

these terms are absorbed

by

the error term in

_(2.1).

The main term on the

right-hand

side of

_(2.24)

with the

_integral

taken over

(201300,00)

equals

which

_by

_(2.25)

reduces to the main term in

_(2.1).

This

_completes

the

_proof

of Theorem 2.1.

3.

_{iY(x, y)}

for small u

In this

_section,

we will _prove Theorem 1.8. We

_begin

with two lemmas

giving

estimates for the

_{approximating}

function

_{A(x, y).}

LEMMA 3.1. _{For any}

_fixed

c &#x3E; 0 and

_{uniformly in y &#x3E;}

_yo(c)

and

1 ~

_yl-E,

, we have

xp(u)

and

Moreover, uniformly

in x &#x3E; 1

_{and y}

&#x3E; 2 we have

Proof.

We may assume that x is not an

integer,

in which case

_{A(x, y)}

is

given by

where

_It}

= t -

_~t~.

_{Using integration by}

_parts,

we obtain

where the last

_integral

is to be

_interpreted

as zero if 0 u 1.

_By

_(1.5)

By

Lemma

_2.2,

this is «

_p(u)ulog(u

+

1)y"(i-E/2)~(u _

_v)

in the _range

1 u

_yl-’.

Hence the last

_integral

in

_(3.4)

is bounded

_by

in this _{range. In} the same _range we have

by Corollary

2.3.

_Inserting

these estimates in

_(3.3)

and

_(3.4)

_gives

_(3.1).

From

_(3.3)

and

_(3.4)

we also obtain

_{A(x, y) &#x3E;}

x ~p(u) - y-"~,

and

_since,

by Corollary

2.3, y-~

for y &#x3E;

yo(c)

and 1 u we see

that the lower bound

_{A(x,y) &#x3E; xp(u)}

holds in this _range. The bound

(3.2)

follows from

_(3.3)

and

_(3.4)

for 0 u

1,

and from

_(3.1)

and and

Corollary

2.3 for 1 u

_~.

For u &#x3E;

_~

we obtain

(3.2)

if we estimate

the last

_integral

in

_(3.4)

as

_{follows, using}

_(1.5),

the

_monotonicity

of _p and

Corollary

2.3,

LEMMA 3.2.

_{Uniformly for}

any E &#x3E;

_0,

_y &#x3E;

_yo(c),

1 _u

yl/3-E

and T &#x3E;

_u5,

we have

where cr = 1 -

~(u)/ log y,

and is

_{defined by}

_(2.15).

Proof.

It suffices to _prove

_(3.5)

when x is not an

integer.

Set

By

Lemma

_3.1,

_Ay(u)

is bounded on _any finite interval in

_{(0, oo),}

and

is

_absolutely

_convergent

for Re s

_{&#x3E; - 3 (log y)~3.}

_By

the definition of

A(y" , y)

and the convolution theorem for

_Laplace

transforms we have

Setting t

=

Y’

and w = 1 ₊

s/ log y,

we obtain

Hence

By

the

_Laplace

inversion formula it follows that

for

_{any u &#x3E;}

1 and Q = 1-

~(u)/ log y

&#x3E; 1

such that x =

y’~

is not a

_positive

integer.

To obtain

_(3.5)

_{(for z g N),}

it therefore remains to show that the

estimate

holds for (T = 1- and T &#x3E;

u5,

since

by

Lemma 2.2 and the

given

bounds on _y and _u,

)

if the constant

_{Yo (f)}

is

_{large enough.}

By

Lemma 2.7 we have for (T = 1 -

ç(u)/logy

and

Irl

&#x3E;

and therefore

Since

₁₍₈₎₁

« for u

&#x3E; 2

and

1,

the second term here is

bounded

_by

_x(7T-1/4,

and hence

_by

the

_right-hand

side of

_(3.7).

_Using

the

_approximate

functional

_equation

for the zeta function in the form

(see,

for

_{example, Corollary}

11.3.5.1 in Tenenbaum

_(1990a)),

we obtain for

the first

_integral

the bound

with

Tn =

_{max(T, n).}

The last term in this

_expression

is of order

«

x~T -2~3

and hence bounded

by

the

_right-hand

side of

_(3.7).

In the first term we

apply

the estimate

which is

_{easily proved by}

an

_{integration by}

_parts,

and

split

the range of

summation into two

_parts

_according

as

T.;I/4

or not. This

yields

the bound

and

_completes

the

_proof

of

_(3.7).

We next _prove an

_analogous

inversion formula for the function

q, ( x, y)

in terms of the

_generating

Dirichlet series

For later _use, we state the result in a somewhat more

general

form than is

with

Proof. Expanding

( s, y)

into a Dirichlet series and

_using

the relation

where

(cf.

Titchmarsh and Heath-Brown

_(1986),

_p.

₆₁₎

_gives

_(3.8)

with

The contribution of the terms with

_I

&#x3E;

1/v~T-

to the series in

(3.10)

is bounded

_{by «}

_{x(7((a,y)/VT,}

and hence

_by

the

_right-hand

side of

(3.9).

It therefore suffices to estimate the

_expression

We have

_trivially

we obtain

since

_by

_(3.11),

= 0 for

Ui.

The

required

estimate now

fol-lows. The next lemma shows tha.t the function

_F(s,

_y)

_closely

_approximates

the function

_{((s, y).}

LEMMA 3.4. For any

given

6 &#x3E; 0 the estimate

holds

_uniformly

in the _range

where

_L,

= = _exp

Proof.

We first note

_that,

in the _range

_(3.13),

since

The sum on the

_right

of

_(3.14)

is

_essentially

a

_prime

number sum which can

be evaluated

_using

_complex

_integration

and

_{Vinogradov’s}

zero-free

_region

in the same _way as in the

analytic

_proof

of the

_prime

number

theorem,

giving

the estimate

for the _range

_(3.13)

-

cf. Tenenbaum

_(1990a),

_p. 419. From

_{(3.14), (3.15),}

Lemma

_2.6,

and the definition of

_{F(s, y),}

it follows that

Integrating

this relation over the

_straight

line

path

from s to 1

_(which

is contained in _{the range}

_(3.13)

if s is in this

_range),

we obtain

The result now

follows,

since

by

Lemma 2.6 and the estimate

which itself follows from a

_strong

form of Wertens’ theorem.

Proof of

Theorem 1.8. Let E &#x3E; 0 and _{x &#x3E; y} with I u

_Le

be

_given.

We

may assume

that y

is

sufhciently large

in terms of E and that u is

strictly

greater

than 1. Set

-1 1

By

Lemma 2.2 we then have in the range _y &#x3E;

_yo(c),

1 u

_Le,

and T &#x3E;

us.

The

_hypotheses

of Lemmas 3.2 and

_3.3,

and those of

Lem-ma 3.4 with s =

T,

a,nd

c/2

in

place

of _c, are therefore

satisfied,

and we obtain

’A 3

with

To _prove the desired estimate

_(1.28),

it thus remains to show that each of the terms

Ri

satisfies

Since

_by

Theorem

_2.1,

Lemma

_2.2,

and Lemma

_3.1,

(3.16)

is

_equivalent

to

To _prove

_(3.17),

we first observe that

Since

T1/4

= _»

_L,

and

₁

x3/4

_{« the term}

_.R1

_{and the}

,E/2

-first term in the definition of

R2

satisfy

the bound in

_(3.17).

_Moreover,

_by

Lemma

_3.4,

Lemma

_2.6,

and the definition of (1 and T we have

which

_implies

_(3.17)

for the second term in the definition of

_R2

in view of the bound

Since

_{I(u+ir, y)}

_I

_{(u, y),}

the same estimates show that the contributions to

_R2

and

_R3

_arising

from the _range

u 5

of the

_integrals

in these terms

also

_satisfy

_(3.17).

The

_remaining

_pa,rts

of

_R2

and

_R3

are bounded

by

respectively,

where

To estimate the

_quantity

_M,

we note

_{that, by}

Lemmas 3.4 and

_2.6,

we

have in the T

The definition of Q and T

implies

that

(1 - 0’)

»

(10gT)-2/3-11

with a

suitable 17

=

77(c)

&#x3E; 0.

_By

standard bounds for the zeta function in

Vino-gradov’s

zero-free

_region

we therefore have

log(T+1)

log y

for 1

_IT

_I

T. It follows that

_(log

_y)2.

This

_implies

the bound

(3.17)

for

_R’.

_By

_(3.18)

the same bound holds for

R’

if u &#x3E;

_{3log LE.}

For

1 ~

_3logL,

we have

so that

_(3.17)

holds in this case as well. This

completes

the

proof

of

Theo-rem 1.8.

4.

_w(x,

_y)

for

_large

u

In this section we will _prove Theorem 1.6 for the

_{range u &#x3E;}

(log

y)4.

In the

_{complementary}

_{range u}

_(log

_y)4,

the result can be deduced from

Theorem

_1.8,

but the

_argument

is somewhat technical and we shall not

present

it here. Given x &#x3E; _y &#x3E;

_2,

we let a =

a(x, y)

be defined as in the

theorem,

we write as usual u =

log x/ log y

and set

LEMMA 4.1. We have

where the

_expression

on the

left of

(4.3)

is to be

_interpreted

as 1

if

a = 1.

Proof.

By

the definition of a we have

For u &#x3E;

_{y/ log y}

this

_implies

0

_{I / log y,}

so that

_{pl -}

1 ~

_{a log p}

for

p

y. The two middle terms in

_(4.4)

are therefore of the same order of

magnitude

in the _{range u} &#x3E;

_{y/ log y.}

This _proves

_(4.1).

The lower bound in

_(4.2)

follows from the fact that a =

y)

is _a

non-increasing

function of u and

_by

_(4.1)

satisfies a »

_{1/log y}

for u =

y/log y.

The _upper bound follows from the

_inequality

which is valid for

_{all y &#x3E;}

2 and u &#x3E; 1 with a

_{sufficiently large}

constant c.

It remains to _prove

_(4.3).

For u &#x3E;

_{y/ log y,}

the

_right-hand

side of

_(4.3)

equals

y/ log y,

and

_by

_(4.1)

the left-hand side is of the same order of

magnitude.

Thus

_(4.3)

holds in this case. To deal with the

remaining

range 1 u

y/ log y,

we use the estimate

which follows

_{by partial}

summation and

_{Chebyshev’s}

_prime

number bounds. In the _range 1 u

_{y/ log y,}

we have a »

_{1/ log y}

by

(4.2),

and it is

_easily

checked that in this case

Hence

_(4.3)

remains valid for 1 u

_{y/ log y.}

We remark that the above

arguments

can be refined to show that the estimate

holds

_uniformly

_y &#x3E;

_2;

see Theorem 2 of Hildebrand &#x26; Tenenbaum

(1986).

The next lemma

_gives

estimates for the functions

_y)

defined in

Theorem 1.6.

LEMMA 4.2. For _any

_fixed

_{positive integer}

k and x &#x3E; _y &#x3E; 2 we have

Proof.

Setting

f (t)

=

1/(et -

1),

_we have

Since for _any

_{integer £}

0 and real t &#x3E;

_0,

it follows that

_(-1)k~k(a,

_y)

is

_positive

and satisfies

Using partial summation,

the

_prime

number

_theorem,

and the bounds

0 a 1 + from Lemma

_4.1,

it is

_{straightforward}

to

_replace

the sum over _p

by

an

integral,

so that

If u &#x3E;

_{y/ log y,}

then a «

_{1/ log y}

by

Lemma

_4.1,

and therefore t’ x 1

and 1 - t-’ x

_{a log t}

_{for 1}

_{t y.} The

_right-hand

side of

_(4.8)

then becomes

by

another

_application

of

_{Lemma 4.1,}

and

_(4.6)

follows for the _range

u &#x3E;

_{y/ log y.}

In the

_remaining

_range 1 ~

_{y/ log y}

we have a y

_{1/ log y}

and

therefore

- ,

for

_q§

t y.

Using

the bound a 1 +

_{0(1/ log y)}

from Lemma

_4.1,

it is

_easily

seen that we _may

replace

the _range of

integration

in

(4.8)

by

the

interval

_~~,

_y]

without

_changing

the order of

_magnitude

of the

_right-hand

side of

_(4.8).

We thus obtain

i

using

the last

_part

of Lemma 4.1. This _proves

_(4.6)

for 1 u

_{y/ log y.}

LEMMA 4.3. For any

fixed

c &#x3E; 0 and

_uniformly

_for

_{y &#x3E;}

2 and u &#x3E;

_(log

_y)4,

we have

’

(4.9)

.

where

_TE(y)

=

exp and c is a

positive

constant.

Proof.

A

_simple

_computation

_gives

with

Thus,

to prove

_(4.9)

it suffices to show that in the _ranges

_lr

₁

_{1/log y}

and

I/log y

Irl

1

TE(y)

we

_{have, respectively,}

Suppose

first that

_Using

the

_{elementary inequality}

1 -

_4x2

_1),

we obtain

If

_(and

then we have

for _{p y,} and hence

which _proves

_(4.10)

in this case. In the

remaining

case a

ITI

1/ log y,

we have

pcx x

1 and

for

_{p y,}

which

_{again implies}

_(4.10).

Assume now tha,t

_{1 / log y}

TE(y).

For bounded values

of y,

(4.11)

holds

_trivially,

and we _may therefore _suppose

that y

is

sufficiently

large

in terms of c. Our

_starting

_point

is the estimate