ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Seria I: PRACE MATEMATYCZNE X I (1968)
ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X I (1968)
M. Ka le c k i (Warszawa)
We prove below the following theorem concerning this sum.
Th eo r em. Let f(y) be a positive continuous and increasing func
tion for у > 1 such that limf(y) — oo. Then
where { } denotes the fractional part of a number, C is a positive constant and z — <p(x) satisfies the equation
P roof. We have
On the sum
oo
X X
— = Cx+0( l ) where
00
Hence
X
d a +0 (1).
1 9 0 M. K a l e c k i
Furthermore
0{z)+0
We now assume that z fulfills the equation z —f(%/z). Since f{x/z) is a decreasing and continuous function of 0 for 0 < x, this equation has for a sufficiently large x a solution z = rp(x) which is unique and satisfies the condition 0<<p(oc)<x. Moreover, bearing in mind that lim/(y)
= 00 we arrive at the conclusion that limę? (a?) = 00 since z = 0(1) con
tradicts z —f
It follows from this that z — o(x); indeed, when z tends to infinity along with x, the ratio xjz must increase to infinity as well.
Therefore
OO
0(9? (ж)) = o(x) and thus
Couollauy. I f f(y) = у , then
X ' where
1
On the sum
Ж Ш)
191у being Euler's constant and z — <p(x) being determined by the conditions z = xjz, z = cp{x) — Vx.
It follows directly that
= a - y x o + o t f o ) from which one may derive the known formula
xlogx-\-(2y — l )x+0{Vx).
