f (t_ it) ~(s) -- s-- 1 s t~+~ dt, (s > O, s#

The distribution of square-full integers

D. S U R Y A N A R A Y A N A and 1%. SITA I~AI%IA CHANDRA_ I~AO Andhra University, Waltair, India

1. Introduction

I t is well-known t h a t a positive integer n is called square-full, if in the canonical representation of n into prime powers each exponent is ~ 2; or equivalently, if each prime factor of n occurs with multiplicity at least two. The integer 1 is also considered to be square-full. Let L denote the set of square-full integers and l(n) denote the characteristic function of the set L, t h a t is, l(n) = 1 or 0 according as n 6 L or n ~ L . L e t L ( x ) denote the enumerative function of the set L, t h a t is, L ( x ) = ~ l(n), where x is a real variable ~ 1.

n<_x

In 1934, P. ErdSs and G. Szekeres (cf. [7], w 2) proved the following asymptotic formula, using elementary methods:

~(3/2)

_ _ xl/2

L ( x ) - - $(3) -[- 0(xl/3). (1.1)

A simple proof of this result has been given later b y A. Sklar [12]. In 1954, P. T. B a t e m a n [1] improved the result (1.1) b y means of the Euler Maclaurin sum 'formula to

where

~(3/2) $(2/3)

X 1/2 - - x l / 3 j [ - O ( x l / 5 ) ,

(1.2)

L ( X ) - - ~(3) + $(2)

co

f (t_ it])

~(s) - - s - - 1 s t ~ + ~ dt, (s > O, s # 1); (1.3)

1

.and he remarked that, b y more delicate methods, it is possible to sharpen the error term in (1.2) to O(x 1/6 log 2 x).

196 D . S U ~ Y A : N A ~ A Y A : b T A A N D 1:~. S I T & I~A_~A O H A N D R A R A O

Let

~(3/2) x 1 / 2 - ~(2/3~) x 1/3. (1.4)

•(z) = L(x) ~(3) ~(2)

In 1958, P. T. B a t e m a n and E. Grosswald (cf. [2], w 5) improved the O-estimate of the error term in (1.2) to a considerable extent b y proving t h a t

d ( x ) = 0 @ 1/6 exp {-- A log 4/7 x(log log x)-3/7}), (1.5) where A is an absolute positive constant. They also remarked (cf. [2], p. 95, lines 30--31) that one can expect to get an estimate of the form A(x) = O(x~), ~ fixed,

< 1/6 if and only if the least upper bound of the real parts of the zeros of the Riemarm Zeta function is less than unity.

In this paper, we further improve the order estimate of d (x). In fact, we prove t h a t A ( x ) = O(xl/6~(x)), where ~(x) = exp {-- A log 3/5 x (log log x)-l/s}, A being a positive constant. Further, on the assumption of the Riemann hypothesis, we prove t h a t :3(x) = O(x(1-+)/(7-12%o(x)), where o~(x) ~-- exp {A log x (log log x)-l}, A being a positive constant, and where 0 is the number which appears in the divisor problem, viz.,

1 = ~(3/2)x 1/2 =Jr ~(2/3)x 1/3 ~ O(x+). (1.6)

a2b 3 <~ x

I t is known that 1/10 < 0 < 2/15. For the lower and upper bounds of 0, we refer respectively to E. Kr~tzel (cf. [9], Satz 7) and H. E. Riehert (eft [10], Satz 2).

We point out t h a t whereas the proof of (1.5) given b y P. T. B a t e m a n and E. Gross- wald [2] is somewhat complicated and m a n y details are missing, the proofs of our results which will be given in w 3 are straightforward and elementary.

A brief historical account of the work done on L(x) b y various earlier authors is given b y E. Cohen (eft [5], w 1). I t should be mentioned t h a t P. ErdSs and G.

Szekeres [7], B. Hronfeck [8], P. T. B a t e m a n and E. Grosswald [2] actually considered k-full integers, t h a t is, integers each of whose prime factors occur with multiplicity at least k, where k is any fixed integer ~ 2. For some other generalizations of the problem, we refer to A. Wintner (eft [15], Section 62), and E. Cohen (cf. [3], [4] and [5]). For weaker order estimates of the error term of the enumerative function of k-full integers, using purely elementary methods, we refer to E. Cohen and K. g. Davis [6].

2. Preliminaries

I n this section we recall some of the known results which are needed in our discussion. Throughout the following, x denotes a real variable > 3. We need the following best known estimate concerning the average of the M6bius function /~(n) obtained b y Arnold Walfisz [14]:

THE DISTRIBUTION OF SQUARE-FULL INTEGERS

LEMlVIA 2.1 (of. [14], Satz 3, p. 191).

M ( x ) = y~ i f ( s ) = o ( x ~ ( x ) ) ,

where

5(x) --- exp {-- A log 3/5 x (log log x)-I/5}, A being a positive constant.

LEMMA 2.2 (cf. [11], L e m m a 2.2). For any s > 1,

~(n) 1 ~/~(x)'~

where

A

Lr, MMA 2.3 (of. [13], T h e o r e m 14--26 (A), p. 316).

I f the Riemann hypothesis is true, then

M ( x ) = ~ /~(n) = O(xl/2(,o(x)), n<:x

~(x) = exp {A log x (log log x)-l}, being a positive constant.

L]~MMA 2.4 (cf. [11], L e m m a 2.6). For any s > 1, if(n) 1

n ~ - + O ( x ' / 2 - ~ o ~ ( x ) ) .

. < ~

~(s)

197

(2.1)

(2.2)

(2.3)

(2.~)

(2.5)

(2.6)

3. Main results

I n this section, we prove t h e results m e n t i o n e d in t h e introduction.

THEO~E~Z 3.1..For x ~ 3,

~(3/2) ~(2/3)

L ( x ) - - - - z '12 + - -

~(3) ~(2)

o(xl/ff~(X)),

vhere ~(x) is given by (2.2).

Proof. We n o t e t h a t a n y square-full integer can be u n i q u e l y r e p r e s e n t e d as n = d268, where 6 is square-free, t h a t is, an integer which is n o t divisible b y t h e square of a n y prime. H e n c e

198 D . S I I L ~ Y A N A ~ A Y A ~ A A N D R. SITA I~A2~IA CItANDI~A ltAO

so t h a t

~(~) = ~ be~(,~)= Y. Z be(e)= 2: be(~),

d Z J a = n dZ6a=n e~f=5 d2e ~ f a = n

L(x) = ~ l(n) = ~ be(e).

n ~_ x d~e6f 3 ~ x

L e t z = x ~/6. F u r t h e r , let 0 < e = ~ ( x ) < 1, where t h e f u n c t i o n e will be s u i t a b l y chosen later.

I f d~er 3 ~_ x, t h e n b o t h e > ~z a n d d2f 3 > ~-6 c a n n o t s i m u l t a n e o u s l y hold good, a n d so we h a v e

L(x) = ~ be(e) q- ~ be(e)-- ~ •e) = S 1-q- S 2 - S a, say. (3.3)

e <_ ~z d2f3 <~ ~ -6 e < ~z

d % 6 f ~ ~ x d~eef ~ < x d~f a < V -~

:Now, b y (1.6),

& = ~ be(e)=Sbe(e) ~ 1

e < ~z e < e z d~f ~ <-- x/e ~

dSe~f a <. x

{

= ~ be(e) ~(3/2) ~ - + ~(2/3) ~ - + 0

e < ~ z

= ~(3/2)x'/~ ~ ~ +

~(2/3)x'/~ <~o~ e--:-

Since, 60 < 12/15 < 1, we h a v e

x ~ ~. j ~ = O(xO(~) '-~o) 1

= o(d-0o~).

e < _ ~ z

H e n c e applying L e m m a 2.2 for s = 2 a n d 3, we h a v e

W e h a v e

f l

& = ~(a/2)x'/:l~-(~ + o

+ r '/3 -]- 0

~(3/2)

~(2/3)

~(3) ~(2)

(ez)2/J

ez / j + : 0 ( e 1 - % )

x ~/~ + O(q-2zt(ez)) + 0(q1-%).

$2= ~ be(e)= ~ ~ be(e)=

~V0<~-, dV,_<o-, ( ~ "1l/,

dZe~f ~ < a e <-- k d ~ a /

(( ^{x I}

=o(x ~/~) ~ d-'/~f-x~,~\~,d~ / /,

d ~ f z <_ e -~

X I/6

)

d ~ f 3 <-- 0-"

(3.~)

THE DISTRIBUTIOI~ OF SQUARE-FULL INTEGERS 199 by (2.1). Since ~(x) is monotonic decreasing and (x/d2f3) 1/6 > qZ,

( ( X /1/6/

~ \ ~ d - ~ / / -<~(Q~) Also,

Hence

X y_ d_,:3 y f_1:

._~

[ /'~)-- 6"~ lj6\

we h a v e

= o ( e - ' .~ d -:!~) = o ( e - ' ( e - 3 ) 1!;) = o(e-~).

d_< ~o -3

S: = o(e-2z(~(ez)).

Also, we have by (2.1) and (1.6),

& = Y :4e) = ~ / 4 e ) Y 1 = O(M(~)~ -~) = O(e-~z~(~)).

e <_ ez e <_ ~z d2f ~ ~ ~s

d2f8 < ~-6

Hence by (3.3), (3.4), (3.5) and (3.6),

r

~OW, w e c h o o s e

~o = e(x) = {~(xl:'~)} ~!~, and write

f(x) = log 3!5 (x ':2) {log log (x1:2)} -1/5 = (1/i2)3/5u3/S(v -- log 12) -1/5, where u = l o g x and v = l o g l o g x .

For v > 2 1 0 g 1 2 , t h a t is, u ~ 144, x ~ e m, we have

v -1/5 < (u -- log 12) -1/5 _~ (v/2) -x/S so t h a t

(1112)3/5u%-'/5 ~_ f(x) :_ (11127/~(112)-'%% -1/5.

We assume without loss of generality t h a t the constant A unity. By (3.8), (2.2) and (3.9), we have

e = e x p { - - Af(x)/3}.

By (3.10), we have (1/3)(1/12)~/5(1/2)-l/su3!Zv-1/5< u/12.

Hence by (3.11), (3.12) and the above

(3.5)

(3.6)

(3.7)

(3.8) (3.9)

(3.1o)

(3.11)

in (2.2) is less t h a n

(3.12)

2 0 0 D . SUI~YANAB, AYA-~q-A A N D l:~. S I T A I:{AMA C]:IANDI:gA I:{AO

e --> exp {-- A(1/3)(1/12)3/5(1/2)-llZu3/Sv -~/5}

> exp -- ~ = exp -- 12 J ' so t h a t ~ > x -1/12. H e n c e

~z > x (lj6)-(1/12) = x U12. (3.13)

Since ~(x) is m o n o t o n i c decreasing, ~(~z) _~ 5(x ljxs) --~ 3 , b y (3.8) a n d so, b y (3.11) a n d (3.12), we h a v e

/ }

~-2d(0z ) ~ ~ _~ exp -- ~ (l/12)3/Su3/Sv-1/5 . (3.14) H e n c e t h e first O-term in (3.7) is equal to O(x ~/6 exp {-- (A/3)(1/12)3/ZuZ/Sv-1/5}), which is O(x ~/6 exp {-- (A(1 -- 60)/3)(1/12)3/Su3/Sv-~/~}), since 0 < 1 - - 60 < 1. B y (3.12) a n d (3.11), we see t h a t t h e second O-term in (3.7) is also of the above order.

Thus i f 3(x) denotes t h e sum of the two O-terms in (3.7), we h a v e

A(x) : O(x 1/6 exp {-- B log 3/; x (log log x-1/5)}), (3.15) where B --- (A(1 -- 60)/3)(1/12) 3/~, a positive constant.

H e n c e Theorem 3.1 follows b y (3.7) a n d (3.15).

T~EOREM 3.2. I f the Riemann hypothesis is true, then the error te~m A(x) in the asymptotic formula for L(x) is O(x(1-~)'(7-1e~)co(x)), where 0 is the number given by (1.6) and co(x) is given by (2.5).

Proof. Following t h e procedure a d o p t e d in T h e o r e m 3.1 using (2.6) instead of (2.3), we get t h a t

A(x) = O(e-~/ez~:Zco(~z)) -~- O(o~-6~ (3.16) Now, choosing ~ = z -1/(7-~2~), we see t h a t 0 < ~ < 1 a n d

~-5/2zl/2 ~ ~1-6~, z _ x(1-o)/(7-12e).

Also, since co(x) is m o n o t o n i c decreasing a n d ~z < z, we have co(Oz) < co(z) < co(x).

H e n c e b y (3.16) a n d t h e above, we h a v e A (x) = O(x (~-~

H e n c e Theorem 3.2 follows.

COnOLLARY 3.1. I f the l~iemann hypothesis is true, then A(x) = O(x~z/~'co(x) ).

THE DISTRIBUTION OF SQUAI:~,:E-~ULL INTEGERS

Proof.

H e n c e t h e c o r o l l a r y f o l l o w s b y T h e o r e m 3.2.

COROLLARY 3.2. J f the R i e m a n n hypothesis i8 true, then A ( x ) = O(xO3m)+~), f o r every e > O.

Proof. T h i s f o l l o w s b y C o r o l l a r y 3.1, s i n c e m(x) =- O(x~), f o r e v e r y s > 0.

201

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Received February 2, 1972 I). S u r y a n a r a y a n a

1%. Sita R a m a C h a n d r a 1%ao D e p a r t m e n t of M a t h e m a t i c s A n d h r a l J n i v e r s i t y W a l t a i r , A.P., I n d i a