Chapter 8 Reciprocals of Additive Functions Restricted to Particular Sequences of Integers

CHAPTER 8 RECIPROCALS OF ADDITIVE FUNCTIONS RESTRICTED TO PARTICULAR SEQUENCES OF INTEGERS

81. Introduction Our attention so far has been focused on sums of reciprocals of functions f taken over all integers n s x for which llf is defined. It seems natural to consider the possibility of evaluating the same sums

when n is restricted to a subsequence of the natural numbers N

.

In

other words, we can consider the sum

where f is additive and

is a subset of N

.

The purpose of this chapter is to estimate (8.1) for those functions f such that f E Sa of Chapter 2 or f is the logarithm of a multiplicative function as in Chapter 3. For f with

R

case, that is, when f E take

EL

and finally

12.

, where

= { n E N :k In}

A

EN .- k l n 1 ,

R

k

2

E

Sa

we shall consider (8.1)

1 is a fixed integer. In the other

(see Definition 8.2), we shall successively = { n E N . -n3(modk)l

= { n E N : (n,k)

11

(where (k,Z)

1

,

.

"Small" additive functions and quotients of additive functions We first consider the class of arithmetical functions defined in

Chapter 2, more precisely, the set of functions Sa , a E R

mi

, of Definition

20 2

DE KONINCK AND A. IVId

J.M.

2.2. Our goal is to estimate

1'

, where

l/f(n)

nsz, ns O(mod k)

and

feSa

k'>l is a fixed integer. In order to do this, we state the following definition. Definition 8.1. Let

and let k E J . If D ( t )

feSa

responding function of Definition 2 . 2 , then for t

E

[O,ll

is the cor-

, Bi(k,t)

is

defined by

x

and A i ( t )

n

(tf(P

U

p-a + t f ( p

at1

+

. . .) D ( t ) t-5

(i-1)

is defined by =

Bi(t)

We shall also write B i ( t )

for B i ( k , t )

Ai(t)

for each i 1,2,...,a t 2

.

The desired estimate for

1'

and Ai

l/f(n)

n%, n-0 (mod k)

for Ai(l)

.

is closely related,

as we shall see, to the behaviour of

where we write n O(k)

instead of n z O(mod k)

.

Therefore, we shall

seek an appropriate factorization of the Dirichlet series (8.2).

This is

REST~ICTICN TO PARTICULARSEQUENCES OF INTEGERS

203

provided by the following r e s u l t . Lemma 8.1.

Let

k

be a multiplicative function such t h a t

m

1 k ( n ) n-' n=l

converges absolutely f o r Re s = o > o

integer, then we have f o r Res

.

If

k

2

1 is a fixed

a > uo

n=l ns

P

The proof is by induction on ~ ( k ,) the number O f d i s t i n c t a prime factors of k If w(k) 1 , then k = p , 1 , and since h is Proof.

.

multiplicative, we have f o r Res

(8.4)

-1

h(n) = 7

n=l n a1 n:O(P1 1

1

m

n=l (n,pl) =1

>

uo

a 1 h(V1 1 -+ a (nPl1Is

c

m

n=l (n,pl) =1

...

a . +I

a.

which proves (8.3) f o r w ( k ) 1 . Assume t h a t ( 8 . 3 ) holds f o r Now l e t k = pal l . . . p ,ar As i n (8.4) we have

.

w ( k ) = r - 1.

204

J.M.

DE KONINCK AND A. I V I e

Defining the multiplicative function 6(n) by a 1 if (n,p, r) 1 6(n) otherwise ,

,

we have, using the induction hypothesis,

c

m

n=l

m = n8

a

nzO(pll . . .p,-a,P- 1)

n

l

m

h(n)

n=l 1

t...)

n8

a

l (wl ...PraP )=I

Substituting this in (8.5),

we obtain the desired result.

The representation of the Dirichlet series ( 8 . 2 ) provided by Lem-

ma 8.1 (with h(n) = tf(n)) for

enables us to derive an asymptotic expansion

1' l/f(n) , which we state as na,nE0(k) Theorem 8.1. Let f e S a

and let k be a fixed positive integer.

Then

(8.6)

1'

nsx n:O(k)

l/f(n)

a Ai a x 1 i=1 (log log x)z

.( (loglogx)atl X

RESTRICTION TO PAFCKULAR SEQUENCES OF INTEGERS

where the Ails Proof.

20 5

a r e defined i n Definition 8.1. Applying Lemma 8.1 with h ( n ) = t f ( n )we obtain f o r

O < t S l

-1

(8.7)

tf(4= -

n = l ns n=O(k)

>

0

, it

(I+-tf (PI t . . . ) -1

PS

plk

Since f E S a Res

n

, and

the products appearing i n (8.7) a r e regular f o r

follows that

t . . . ) -1 t-t-

P

nsx n=O(k)

where D ( t )

P2

i s defined i n Definition 2 . 2 , and

R(z, t) = uniformly f o r

0
5

1

.

O(Z

x)

Proceeding a s i n the proof of Theorem 2.4 we

obtain Theorem 8.1.

Applying Theorem 8.1 t o the functions observing t h a t

w

and n of

Su

, and

J.M. DE KONINCK AND A. IVId

206

n[lt$C.+E P Ik

t . . . )-1

n -,[

$Pa)

t...)

1

Patl

Pal Ik

P

t

,

F1 t=l

we obtain the following c o r o l l a r i e s . Corollary 8.1.

1’

n a n=O(k) where a;

k

and M are fixed positive integers, then

M

a: i=l (log log x)

1

l/w(n)

1

If

, and

X

O( (loglogx)Mtl

the remaining constants, which depend on k

, are

com-

putable. Corollary 8.2.

where b;

1

, and

If

k

and M are fixed positive integers, then

t h e remaining constants, which depend i n k

, are

com-

putable.

Following the method of proof i n Chapter 4 we s h a l l now estimate sums (over

nzO(modk)) of quotients of additive functions.

Theorem 8.2. f o r a l l primes p

Let g and f be two additive functions such that

and a l l integers r t 1

20 7

RESTRICTION M PAFWCULAR SEQUENCES OF INTEGERS

where C1 and

C2

a r e two positive absolute constants.

Then, f o r

k>l

a fixed integer, (8.8)

1'

$$- =

nn=O(k)

k

Bx loglogz

t

t

o

[

X

(log log x)2

where B is a computable constant depending on F,g

, we

special case g = R , f = u

(8.9)

m-r;

1'

nlz:

Q(n)

+

, and

k

.

In the

have M

1

X

n=O(k)

where bf21

is an a r b i t r a r y but fixed integer, and the

t a b l e constants depending on k Proof. It( s 1

, and

hi's

a r e compu-

.

Applying Lemma 8 . 1 t o

h ( n ) = t g ( nu)f ( n )

where

tE E

u ~ ( 0 , l I we obtain

Using (4.7) and the f a c t t h a t the products i n (8.10) a r e regular

,

208

J.M. DE KONINCK AND A. IVId

functions of

s

for Res > 1 - E

, we obtain uniformly for

It I < 1

, Iu I

5

1

n:O(k)

is absolutely and uniformly convergent for Res > 1/2 .

where K(t,u;s)

Following the proof of Theorem 4.1 we obtain (8.8) and (8.9),

by observing

that

The reader might wonder why we did not try to obtain estimates for the more complex sums

for ( k , Z ) = 1

.

What we can obtain is a "reasonable" representation for

the series

(8.13)

where f is additive and O < t < 1 X1' X2'".'

x9(k)

.

Indeed, let (k,Z)

be the characters modk

, where x1

=

1 and let

is the principal

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS character (xl(n)

=

1 if (n,k) = 1 and zero otherwise).

If

209 6o

is the

m

abscissa of absolute convergence for ve, then for Re s

n-2 ( k )

1 h ( n ) n-’ n=l

and h is multiplicati-

=6> 6

II

.

But

hence

A

close analysis of relation (8.14) with h ( n ) replaced by

tf(n)

shows that it is not possible to factor out from (8.14) an expression of the form ( ~ ( s ) ,) which ~ would allow us to use the powerful Lemma 2.1. Nevertheless a careful study of the function

where x

f

x1 , might lead to an estimate for the intricate expression

210

J.M.

DE KONINCK AND

A. IVId

where for Re s > 1

is the L-series associated with x ( n )

, and

G(t,s;X)

is absolutely and

uniformly convergent for Re s > 1/2 . The problem is then transformed into the problem of estimating

which is very difficult.

93. Reciprocals of logarithms of multiplicative functions

The arithmetical functions defined by logarithms of positive multiplicative functions, which have been already studied in Chapter 3 , are more easily handled than other additive functions. Indeed, the fact that the estimation of sums of reciprocals of these functions, even when n is restricted to certain special sequences of integers, does not require the use of the powerful Lemma 2.1 allows us to somewhat enlarge the scope of our results. The first attempt to study sums of reciprocals of logarithms of a multiplicative function restricted to a congruence class of integers was made by J . M . Tourigny. He proved that, given a fixed prime p , preassigned positive integer M ,

and any

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS

where al

l/po

ding on p ,

.

, and

the remaining ails

Here, of course,

a

211

are computable constants depen-

denotes the sum of divisors function.

In this section, our ultimate goal is to provide a general estimate for

(8.17)

, and

1 or k

where ( k , Z )

f belongs to a certain large class of po-

sitive multiplicative functions. We shall proceed in two steps. First we extend estimate (8.16) to the family

5

of Definition 3.1, giving way rather easily to a precise

estimation of (8.17) for n-O(modpo)

Then, considering a larger class

, we prove several identities concerning generating functions of functions f E & restric-

of functions

& and taking

.

(k,Z) = 1 or

(k,Z) = k

ted to certain congruence classes. Finally we deduce asymptotic expansions for (8.17). Our first result is as follows: Theorem 8 . 3 . integer. If f.

where, for -l/v

Let p ,

$ , then

5

t

5

0

,

be a fixed prime and M a fixed positive

J.M. DE KONINCK AND A. IVI6

212

where ht (n) is defined by (3.8). Proof.

In order t o prove (8.18), we define the function f o ( n )

a s follows: nzO(modpo)

f(n)

if

0

otherwise

fo(n)

,

.

Then

and

(8.19)

To see that (8.19) holds we note that the above sum is zero whenever p o l n by the definition of 9(n) by 9(n) = ( f ( n )

a

a

dlmpoO,PoOI Id

f,

.

-a t

1

Assume n = *

poem a

with

Then

a

a -1

d lmpO0'Po0

I Id

(m,po)

1

, and

define

213

RESTRI(JTI0N TO PARTICULAR SEQUENCES OF INTEGERS

from the multiplicativity of h,

.

Using the method of proof of Theorem 3.1, we have from (8.19) that

which is similar to the crucial relation (3.14) obtained in the proof. Now proceeding with (8.20) as was done with (3.14), we finally obtain (8.18).

We now enlarge the scope of Theorem 8.3 by considering a class of

.

functions somewhat larger than the class belonging to this new class

& ,

defined below, our sums will be over

those integers n s x which satisfy n-Z(modk) We begin by defining

k

t

5

ht(n)

np

P In

>

0 that for every

(8.21)

where a(n)

.

be the set of all positive mltiplicati-

ve functions f such that for some a 5

when ( k , Z ) = 1 or k

& .

Definition 8.2. Let

formly for -l/a

Then for certain functions

, and

ht(n)

<<

0 and some fl E

>

<

0

we have uni-

0

(a(n))BnE

is the multiplicative function defined by

J.M.

214

DE KONINCK AND A. IVId

2

I t i s obvious t h a t

& , where

i s the class of multi-

plicative functions defined i n Definition 3.1, since from (3.11) it can be seen t h a t (8.21) is s a t i s f i e d .

8

I t may also be shown t h a t

is the class of functions defined i n Definition 3.2.

8 If

2

fE

, where B ,

then for a = 1 , B -1 we have

ht@

3

-

1-

-1

( ~ + Q ~ + , .~. . P + aj J j p

= O ( l t ) p-l)

for -l/a

2

t h a t for

-j t

and

since

-j+llt

s cp-1

t s 0 and f o r some suitable C > 0

It1 s l / a

-1

- ( l + a l , j - l ~ +...+uj-l,j-lp

.

Here we used the f a c t

1x1 s 1/3

..

t t ( t - 1 ) . (t-ntl) (n) = n!

and

for p

2po

, where X

>0

is the constant appearing i n Definition 3.2.

thus obtain for some suitable C > 0

since , t r i v i a l l y

We

21 5

R E S T R I f l I O N TO PARTICULAR SEQUENCES OF INTEGERS

Before proceeding t o estimate (8.17) f o r c e r t a i n functions f E %,! we give a general i d e n t i t y f o r the generating function of any multiplica-

t i v e function f r e s t r i c t e d t o a congruence class f o r which t h e abscissa m

of convergence

u

1 f ( n ) n-' n=l

of

0

is f i n i t e .

Let f be a multiplicative function such that

Lemna 8.2. m

1 f(n)

converges absolutely f o r Res > u

n-'

then

n=l

m

n=l

, k'

where d = (k,Z) modk'

k/d

, I'

, and

= Z/d

the x's a r e t h e characters

. Proof.

If

(k,Z) = k (k, I )

becomes (8.3), while if

, then

n i l @ ) means n-O(k)

1 we have

d=1

, and

, and

(8.22)

(8.22) follows

from (8.14). Lemna 8.3. Res>l

and - l / a

Let f E 5

t

5

0

and l e t

( k , 2) = 1 o r k

.

Then f o r

,

216

J.M. DE KONINCK

AND A. IVId

where (8.24)

F(s,k,t)

p

=

t...)

-2a

2

t Y L - d mx t 1 ( P k'

cf(p)p

t

t . . .)

-1

P 2s

PS

-(atl)a t

at1 t

where d = ( k , l ) mod k

-1

k ' = k/d

, Z'

= Z/d

(f@

P(

and the

t...)

a) tP1 ) s

,

x's are t h e characters

. Since f i s positive and -l/a

Proof. 0 s ( f ( n )n-a)t

I,

1

.

t

5

0

we have t h a t

Thus the proof follows immediately from Lemna 8.2

with f ( n ) replaced by

(.f(n) n-")

I,

Cf(n)n-a)t

,

since the non-negativity of

implies the non-vanishing of t h e s e r i e s

217

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS f o r Re s

2

1 , and assures absolute and uniform convergence f o r

of t h e Dirichlet s e r i e s generated by

Re s > 1

( f ( n ) n-")*, - l / a I t 2 0

.

We a r e now ready t o employ our basic method which was outlined i n Chapter 2 and was used i n Chapter 3 f o r summing reciprocals of logarithms of multiplicative functions.

1

mula f o r

We shall f i r s t e s t a b l i s h an asymptotic for-

, where f

belongs t o a subclass of

Let

&

ft(n)

&x, n Z ( k )

&,

, defi-

ned by Definition 8.3.

(8.25)

df(p)p-a)t

E

, where

[-A,O]

We c l e a r l y have

, then

and some 6

>

0

1 t O(p-&)

i n a r b i t r a r y but fixed, and

A >0

except possibly f o r O(xE)

fE

be t h e class of functions be-

such t h a t f o r every prime p

longing t o

for t

f

integers

nsr

5

4.

u

5

f o r which f ( n ) = 1 , In order t o see t h i s , l e t

from (3.4) we have

1- p - 8 s (f(p) p - a p s (1 - p - B ) - 1 = 1 O ( f 8 )

,

which implies (f(pj p T t =

{

(fb)P -"1l l q t / y

=

(l+O@-B))t/y = l t O ( p - 8 )

since t belongs t o a fixed i n t e r v a l .

From ( 3 . 4 ) we have f ( n )

, >>

n

a/2

.

21 8

J . M . DE KONINCK AND A. IVIi

Thus, f e

h1.

Similarly, using (3.20),

L e m 8.4. k

Let f~

.&,

and - l / a

&

5

A1 ,

5

t

5

0

.

If (k,Z) = 1 o r

, then

(8.27)

1

ft(n) = G(l,k,t) zattl attl nsx n-Z(k)

atG(l,k,t) + R ( 0 ) attl

where G is defined by

where d

(8.29)

(k,Z)

,k

k'ld

, and o(z l t a t - p 1

R(a,t)

uniformly in t for some fixed

p >

0

.

Proof. From the definition of ht(n) we have

(8.30)

which in view of (8.23) implies

Y

219

RESTRICTION M PARTICULAR SEQUENCES OF 1NTEC;ERS

m

(8.32)

B(s,k,t)

and f o r every

E >

1 b(n,k,t) n-' n=l

,

0

1 b(n,k,t)

(8.33)

=

n-

,

O(Z~-~'€) t ) '~'Z(O

since uniformly i n t

To see that (8.34) holds, we note t h a t using (8.25) we have f o r Res>l

n(1+X ( p ) f S

+0(f6) p - s + X*(p) ( p - 2 Q f ( p 2 ) ) t p - 2 s +.

P

=

n

P

(1 - x@) p - s ) -l

L(s,x)

t O(p-6)

P

n (l+O(p-6)p-St...)

P where

r[ ( 1 - x@) p-) ( 1 t x(p> p - s

.. 1

L(s,x)

C(s,x>

>

p - s t . . .)

J . M . DE KONINCK

2 20

AND

A. IVId

and

is regular f o r Re s > max (1- 6,1/2) for X = X

1

and k

, which

implies then (8.34), since

fixed,

A l l the products i n (8.24) of the type

are non-vanishing f o r Re s = 1 a regular function of

s

.

This means t h a t i n (8.24)

f o r Res > 1- p f o r some fixed

F(s,k,t) p >

0 , Using

a convolution argument we then i n f e r from (8.31) and (8.33) that

The above equation gives

(8.36)

is

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS

I t i s now c l e a r why a condition such as (8.21) was needed. recalling t h a t from Theorem 1.4 we have f o r every

E >

221

Indeed

0

we obtain from (8.21)

f o r some

E~

> 0

.

Using p a r t i a l sumnation t o estimate

we obtain from (8.36)

which i n view of remarks made e a r l i e r gives

(8.40)

1

n ~ xn-2 , (k)

( f ( n ) n-’)*

= xG(lyk,t)

t

0(x1-’)

.

P a r t i a l sumnation f i n a l l y gives (8.29) f o r some fixed p > 0 which is not necessarily the s a m a t each stage of the proof.

J.M. DE KONINCK AND A. IVId

222

We are now ready to establish an asymptotic formula f o r the general

sum (8.17). Theorem 8.3.

Let f e

A1 , and l e t

1 or k

(k.,Z)

.

For an

arbitrary but fixed integer M z l we have

n-Z (k) where al = l / k

, and more

generally

a

(-1)

j

where E ( t ) = G(l,k,t)/(at

t

1)

j-1

E

(j-1)

(0)

, G(l,k,t)

,

being defined by the function

appearingin the statement of Lemma 8.4. Proof.

We make use of (8.26), which bounds f ( n ) away from unity,

except f o r O(xE) integers n s x nsx

satisfying f ( n ) 2 2

. Since

.

Let

c'

denote summation over those

f ( n ) >> naI2

2

for n z n

0

, we

for c > a

Since f ( n )

>>

na/'

for a l l n

, and

f(n) 2 2

for n z n o

,

have

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS

1 f ( n ) dt= n~~:,nZ(k)

(8.44) -l/c

1

( E ( t ) xa t t l t a t E ( t ) +O(xl t a t - p

223

1) d t ,

-l/c

where E ( t ) = G(l,k,t)/(at t 1 )

.

Integration by p a r t s gives

Since E ( M ) ( t ) xat

O(1)

for t E Cl/c,Ol

, the

last above integral

is bounded, and, moreover, f o r O s i S M we have

X

di)(-l/c) x -a/c

<<

(a log xli

With uj

(8.45)

1

1-a/c

-

(l0gx)i

, it

defined by a = (-1) j-1 E (j-1) (0)

follows t h a t

j

E(t)x

attl

-l/c

dt

x

M

a

1

X

i = l ( a logx)

Furthermore

a t E ( t ) d t = O(1)

(8.46) -l/c

,

i

) dt

O(xl-')

,

-l/c

and Theorem 8 . 3 follows from (8.42)

,

( 8 . 4 3 ) , ( 8 . 4 4 ) , ( 8 . 4 5 ) , and ( 8 . 4 6 ) .

J . M . DE KONINCK

2 24

AND

A. IVId

Before finishing t h i s chapter, we would l i k e t o consider the sum

when f

E

k,

.

Using Lemna 8.4 and the f a c t t h a t

does not depend on 2

for

(k, 2 ) = 1

- $ ( k ) G(l,k,t) z a t t l

, we

of (8.28)

obtain uniformly f o r - l / a s t s 0

4 ( k ) a t G(l,k,t) at t 1

att 1

G(l,k,t)

O(xltat-p

1 -

Using the method developed i n the proof of Theorem 8.3 we obtain then the following Theorem 8.4.

Let f

E

A

and l e t k

2

1 be a fixed integer.

For an a r b i t r a r y but fixed integer M 2 1 we have

(8.49)

1'

nsx

where al = l / k

i

X

i=1 (alogz)'

(n, k) =1

, and,

more generally, a

= (-1)

,i

where E(t)

a

M l/logf(n) = @ ( k ) z1

G(l,k,t)/(at

t

1)

j-1

E

(j-1)

, where

appears i n the statement of L m a 8.4.

(0)

9

G(l,k,t)

is the function which

225

RESTRICI'ION TO PARTICULAR SEQUENCES OF INTEGERS NOTES

Only the most elementary facts concerning characters and L-series are used i n t h i s chapter.

Two of these are the orthogonality relations $(k)

c

(8.50)

x(n) = n(mod k )

and

(

$(k)

if

x=xl

if

x'xl

if

n-l(mdk)

Y

a proof of which can be found i n standard works such as Chandrasekharan C11 or Prachar C 11. Foxmula (8.9) was obtained by J . M . Tourigny in his Master's thesis (Universit6 Laval, W b e c , 1975).

Theorem 8.2 seems t o be new, while

Theorem 8.3 generalizes ThBorhme 4 of A. Mercier C11.

The identities in-

volving Dirichlet series that were used i n this chapter are also from Mercier's paper, which continues the work of De Koninck and Mercier C11. Identities involving Dirichlet series with multiplicative coefficients, characters, subseries etc., were also investigated by T.M. Apostol i n [21,

Cal and C4l. I t is possible, of course, t o investigate when

&

chapter.

is a subset of

l/f(nl

which differs from the ones used i n this

Such an example has already been given i n Chapter 2 .

an estimate for

(8.52)

W

1'

nsx, ne p

There,

J.M.

226

DE KONINCK AND A. IVI6

was obtained i n Theorem 2.9, where integers.

denotes t h e s e t of square-free

Although generalizations of (8.52), obtained by replacing

w

by other additive functions, are c l e a r l y possible, i n general t h e estima-

1'

t i o n of

6 x ,nE R

l/f(n)

fl

i s d i f f i c u l t , even i f

gers whose d i s t r i b u t i o n is well-known, such as

8

consists of inte-

= { p - 1 :p

i s prime)

.

= {n:n=k2t12,k,Z~~u{O)}

or

I n Lemnas 8.3 and 8.4 one can also take t a E

with

- l / a s Re t s 0

This would allow us t o d i f f e r e n t i a t e (8.27) with respect t o t take t 0

1

nsz

, thereby

logf(n)

.

providing an estimate f o r

1 logf(n) nsk n=l( k )

This was done by A. Mercier C11

, who

and then and

obtained f o r

(n, k) =1 example

n-l (k)

1 lOgo(n)= q z l o g x m (n, k ) =1

where p ,

tants:

t

Apx+O(log2z)

i s a fixed prime, and A1 , A 2

,

a r e two e x p l i c i t l y given cons-

.

RESTRICTION TO PARTICULAR SEQUENCES OF INTEGERS

227

Using the methods of p r o b a b i l i s t i c number theory outlined i n Chapt e r 3 and t h e results on d i s t r i b u t i o n problems of arithmetical functions obtained by Galambos (C3l

, C41)

i t i s possible t o give an estimate of t h e

type

where f(n)

is "close" t o

( 3 . 3 3 ) , and where

log l o g n

i n the sense implied by r e l a t i o n

is a set of positive integers.

However, i n t h i s

case, a much weaker e r r o r term than the one i n (3.39) is obtained.