Arch. Math., Vol. 55, 450-461 (1990)
0003-889X/90/5505-0450 $ 3.90/0 9 1990 Birkh/iuser Verlag, Basel
Continuity module of the distribution of additive functions related to the largest prime factors of integers By J. M. DE KONINCK,I. KATAIand A. MERCIER
1. Introduction. For an integer n > 1, let p (n) and P (n) denote the smallest and the largest prime factor of n, respectively. The letters c, q , c 2. . . . denote suitable positive constants not necessarily the same at every occurence. For some ~ > 0 let (1.1)
f(n) = E (logq) ~, qln
where the sum runs over the prime divisors of n, (1.2)
v~(n) dr _ _ 1
f (n),
(log x) ~ (1.3)
T (n) dr
f (n) (log P(n)) ~"
In our previous paper [1] we proved that both v~(n) (n < x) and T(n) have limit distributions. Let (1.4)
F~(y) = 1_ #e {n < x: v~(n) < y}, X
(1.5) (1.6)
F(y) = lim Fx(y), G~(y) = 1 ~ {n 1 by (1.8)
1
lim - 7t(x, x TM)= 0 (t), x~oo
X
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where hU(x, y) stands for the number of integers n up to x Satisfying the condition P (n) < y. It is known that 0 is a decreasing function, that (1.9)
0 (t)
exp
[(
t log t + log log t
) O(lo+)1
1
log log t + log r
'
as t --* oe and furthermore that (1.10)
7~(x,x 1/') = xo(t) + O(x/logx)
holds as x ~ ~ uniformly for all t varying in a bounded interval (see [1]). The continuity modules of F and G, that is
Qv(h) = max (F(y + h) - F(y)) Y
QG(h) = max (G(y + h) - G(y)) Y
will be treated here. We shall provide (mainly) upper bounds for Qv(h) and Qo(h), where 0 < h < 1, for various ranges of a. Hence the results established in the following sections may be outlined as follows: let 0 < h < 1, then 1/2
Qv(h)
=1 if,=l, /" 1'~+ 1 i_L ~ ~log ~) h ~ if ~ > 1,
and
= log(l + h)
if ~ = 1, 1
Qo(h)
1. Let
(4.1)
E(y) ~f F~(y + h) - F~(y),
where h = 6=. Choose a fixed y > ~ . Let ~ v~,~(n) E [y, y + h]. It is clear that card(~-~)
9 ~(y)
(x-~
be the set of integers n __ x a. Then c a r d ( ~ \ ~ * ) = o(x) (x ~ 0 0 ) . For a general n, let Pl > P2 > --- > Pr be the set of all prime divisors greater than x ~ We shall write nl = PiP2 " " P r ,
M = P2 ' " Pr.
Let us estimate card(9~*). Since y > 6~, then clearly n 1 = 1 cannot occur. N o w let ~ > be fixed. Then (4.2)
card(J~*) =< ~ gJ ( M - - ~ ), x ~ + ~ ( x , x ~ ) = Y + ~ ( x , x ~ ) ,
J.M. DE KONINCK,I. KA.TAIand A. MERCIER
454
ARCH.MATH.
where in 32 we sum only over those n 1 e ~-~* for which Pl > xe. We shall now make use of the inequality (4.3)
~ ( x , x a) ~ x exp
(c) -~
valid uniformly in the range 0 < 2 < 1. To estimate 32 on the right hand side of (4.2), we shall distinguish between two cases, namely: (A)
y > 8 ~ ~,
(a)
y < 8 ~ ~.
C a s e (A). Since for n t c ~-~*, we have (4.4)
v~,o(n~)
= (l~ \logx}
+v~,o(M)e[y,y+h),
it follows that, for a given M, there exists Pt > x~ only if v~, o(M) < y + h - ~ . We shall write ~ as 321 + 322, where in 321, we have the restriction v~,~(M) __ 4~" and
xr 1, then /
1\~ +l
Clogs)
h
1-!
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ARCH. MATH.
5. Estimation of Qr(h) in the case 9 < 1. A general theorem of I. Z. Ruzsa immediately implies the following Theorem 2 (Ruzsa [2]). Let ~ < 1. Then 1
QF(h) ~ log 1/5. Assume n o w that 0 < tc < 1. T h e n
w(x,~)= x~ 0 be given. Then the density 1 of the integers n satisfying y __ 1. Let y > 1 and 0 < h < 1 be fixed. We argue similarly as in Section 6. For E(y,h) = G(y + h) - G(y), we have (8.1)
x E ( y , h ) N x exp +
~2
+ J~2 x card (~'~, 1,) +
p'Zl
h. But Theorem 3 gives that the same is true if 1 < y < I + h. Hence we have
Theorem 6. I f ct > 2, then
clh l/~ ~ Q~(h) < c2h l/~(log l/h) ~+1, for 0 < h < 1, with suitable positive constants that may depend on ~.
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References [1] J.M. DE KONINCK, I. K~TAI and A. MERCIER,Additive functions monotonic on the set of primes. Acta Arith., to appear. [2] I.Z. RUZSA, On the concentration of additive functions. Acta. Math. Acad. Sci. Hung. 35, 215-232 (1980). Eingegangen am 24. 4. 1989 Anschriften der Autoren: Jean-Marie De Koninck D6partement de Math6matiques Universit~ Laval Qu6bec, G1K 7P4 Canada Armel Mercier D6partement de Math6matiques Universit6 du Qubbec Chicoutimi, G7H 2B1 Canada
Imre Khtai E6tv6s Lor~md University Computer Center 1117 Budapest, Bogd/mfy u. 10/B Hungary
