2 - 07

COROLLARY 1.- If y and u tend to infinity, we have

In particular

Y ( x . y ) ^x°^c ( a . y } ( 2TTu ( l + ^)r^/^log(l+J^))"^

Y ( x , y ) ^ - J^ - - - - ^i° : . iXi - - - - - - - - - , (if y/logx-»co} .

y2nu log(y/log X)

and

^^u^y ) ^Jf : li^Ll^ , (if y/iogx^o) .

/ 2TTy / log y

This shows that the behaviour of У(х, y) is quite différente according as y/log X is small or large. This phenomenon has already been stressed by de Bruijn [4] . It is due to the fact that for y ^y ( e) and ky^(l-e) log x , we have by the prime number theorem

( П P) ^ X

so that the numbers that do not have a prime factorization with exceptionally

high powers contribute little to Y(x,y) - for instance the squarefree numbers

l / k contribute at most x . This feature does not occur when y/log x is large.

When we combine Theorem 1 with part (ii) of Theorem 2, we derive a smooth approximation fo Y(x, y) which in the range

5 /З + E ^

x^2 , (loglog X) <logy^yiogx

is as sharp as de Bruijn*s estimate (1.5). Apart from the upper restriction for y , this range coincides with (1.7) where (1.5) is known to be valid. For larger y , the estimates (2,3) and (2. 6) together are less precise than (1,5) because of the error term 0(l/u) in (2.3).

The approximations (1.5) and (2. 6) are of quite different type. Equalizing

/ 1/2

the two expressions in the case u = (log 2y) , say, we obtain the following

result .